Applications of Poisson Geometry to Physical Problems

Applications of P oisson Geometry to Ph ysical Problems Summer Sc ho ol and Conference on P oisson Geometry ICTP , T rieste, Italy , 4-22 July 2005 Prof Darryl D. Holm d.holm@imp erial.ac.uk dholm@lanl.go v August 2, 2007 1 D. D. Holm Imp erial College London Applications of P oisson Geometry 2 Con ten ts 1 In tro duction 7 1.1 Road map for the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Motiv ation for the geometric approac h . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Review Newton, Lagrange & Hamilton 10 2.1 Diﬀeren tial forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Handout on exterior calculus, symplectic forms and Poincar ´ e’s theorem in higher dimensions 14 4 F ermat’s theorem in geometrical ray optics 16 4.1 F ermat’s principle: Ra ys take paths of least optical length . . . . . . . . . . . . . . . . 16 4.2 Axisymmetric, translation in v arian t materials . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 The P etzv al in v arian t and its Poisson brac k et relations . . . . . . . . . . . . . . . . . . 18 4.4 R 3 P oisson brack et for ra y optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.5 Recognition of the Lie-P oisson brack et for ra y optics . . . . . . . . . . . . . . . . . . . 18 5 Geometrical Structure of Classical Mec hanics 19 5.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Motion: T angen t V ectors and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6 Deriv ativ es of diﬀerentiable maps – the tangent lift 24 6.1 Summary remarks ab out deriv atives on manifolds . . . . . . . . . . . . . . . . . . . . . 25 7 Lie groups and Lie algebras 26 7.1 Matrix Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.2 Deﬁning Matrix Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 7.3 Examples of matrix Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.4 Lie group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.5 Examples: S O (3), S E (3), etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8 Lifted Actions 35 9 Handout: The Lie Deriv ative and the Jacobi-Lie Brack et 39 10 Handout: Summary of Euler’s equations for incompressible ﬂo w 41 11 Lie group action on its tangen t bundle 46 12 Lie algebras as vector ﬁelds 48 13 Lagrangian and Hamiltonian F orm ulations 48 13.1 Newton’s equations for particle motion in Euclidean space . . . . . . . . . . . . . . . . 48 13.2 Equiv alence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 14 Hamilton’s principle on manifolds 53 15 Summary Handout for Diﬀerential F orms 55 D. D. Holm Imp erial College London Applications of P oisson Geometry 3 16 Euler-Lagrange equations of manifolds 59 16.1 Lagrangian vector ﬁelds and conserv ation la ws . . . . . . . . . . . . . . . . . . . . . . 60 16.2 Equiv alence of dynamics for hyperregular Lagrangians and Hamiltonians . . . . . . . . 61 16.3 The classic Euler-Lagrange example: Geodesic ﬂow . . . . . . . . . . . . . . . . . . . . 62 16.4 Cov arian t deriv ativ e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 17 The rigid bo dy in three dimensions 67 17.1 Hamilton’s principle for rigid b ody motion on T SO(3) . . . . . . . . . . . . . . . . . . 68 17.2 Hamiltonian F orm of rigid b ody motion. . . . . . . . . . . . . . . . . . . . . . . . . . . 71 17.3 Lie-Poisson Hamiltonian form ulation of rigid b o dy dynamics. . . . . . . . . . . . . . . 72 17.4 R 3 P oisson brack et. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 18 Momentum maps 73 18.1 Hamiltonian system s on P oisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 74 18.2 Inﬁnitesimal inv ariance under Hamiltonian v ector ﬁelds . . . . . . . . . . . . . . . . . 75 18.3 Deﬁning Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 19 Quick summary for momentum maps 82 19.1 Deﬁnition, History and Ov erview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 20 Rigid b o dy equations on SO( n ) 83 20.1 Implications of left in v ariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 21 Manako v’s form ulation of the S O (4) rigid b ody 85 22 F ree ellipsoidal motion on GL( n ) 86 22.1 Polar decomposition of free motion on GL + ( n, R ) . . . . . . . . . . . . . . . . . . . . . 87 22.2 Euler-Poincar ´ e dynamics of free Riemann ellipsoids . . . . . . . . . . . . . . . . . . . . 88 22.3 Left and righ t momentum maps: Angular momen tum versus circulation . . . . . . . . 89 22.4 V ector represen tation of free Riemann ellipsoids in 3D . . . . . . . . . . . . . . . . . . 90 23 Heavy top equations 92 23.1 Introduction and deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 23.2 Heavy top action principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 23.3 Lie-Poisson brac kets and momen tum maps. . . . . . . . . . . . . . . . . . . . . . . . . 95 23.4 The hea vy top Lie-Poisson brac kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 23.5 The hea vy top formulation b y the Kaluza-Klein construction . . . . . . . . . . . . . . 96 24 Euler-Poincar ´ e (EP) reduction theorem 97 24.1 W e w ere already sp eaking prose (EP) . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 24.2 Euler-Poincar ´ e Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 24.3 Reduced Legendre transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 25 EPDiﬀ: the Euler-P oincar´ e equation on the diﬀeomorphisms 101 25.1 The n − dimensional EPDiﬀ e quation and its properties . . . . . . . . . . . . . . . . . . 101 25.2 Deriv ation of the n − dimensional EPDiﬀ equation as geo desic ﬂo w . . . . . . . . . . . 103 26 EPDiﬀ: the Euler-P oincar´ e equation on the diﬀeomorphisms 104 26.1 Pulsons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 26.2 Peak ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 D. D. Holm Imp erial College London Applications of P oisson Geometry 4 27 Diﬀeons – singular momen tum solutions of the EPDiﬀ equation for geo desic motion in higher dimensions 108 27.1 n − dimensional EPDiﬀ e quation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 27.2 Diﬀeons: n − dimensional Analogs of Pulsons for the EPDiﬀ equation . . . . . . . . . . 109 27.2.1 Canonical Hamiltonian dynamics of diﬀeon momen tum ﬁlaments in R n . . . . 110 28 Singular solution momen tum map J Sing for diﬀeons 110 29 The Geometry of the Momen tum Map 115 29.1 Coadjoint Orbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 29.2 The Momen tum map J S and the Kelvin circulation theorem. . . . . . . . . . . . . . . 115 29.3 Brief s ummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 30 The Euler-P oincar´ e framework: ﬂuids ` a la [HoMaRa1998a] 117 30.1 Corollary of the EP theorem: the Kelvin-No ether circulation theorem . . . . . . . . . 123 31 Euler–Poincar ´ e theorem & GFD (geoph ysical ﬂuid dynamics) 124 31.1 V ariational F ormulae in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 124 31.2 Euler–Poincar ´ e framew ork for GFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 31.3 Euler’s Equations for a Rotating Stratiﬁed Ideal Incompressible Fluid . . . . . . . . . 126 32 Hamilton-Poincar ´ e reduction and Lie-P oisson equations 127 33 Two applications 131 33.1 The Vlaso v equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 33.2 Ideal barotropic compressible ﬂuids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 33.3 Euler’s equations for ideal incompressible ﬂuid motion . . . . . . . . . . . . . . . . . . 133 D. D. Holm Imp erial College London Applications of P oisson Geometry 5 Preface These b eing lecture notes for a summer school, one should not seek original material in them. Rather, the most one could hop e to ﬁnd would b e the insigh t arising from incorporating a uniﬁed approac h (based on reduction b y symmetry of Hamilton’s principle) with some nov el applications. I hope the reader will ﬁnd insight in the lecture notes, whic h are meant to be informal, more like stepping stones than a prop er path. Man y excellent encyclop edic texts ha v e already b een published on the foundations of this sub ject and its links to symplectic and Poisson geometry . See, for example, [AbMa1978], [Ar1979], [GuSt1984], [JoSa98], [LiMa1987], [MaRa1994], [McSa1995] and man y more. In fact, the scop e encompassed by the mo dern literature on this sub ject is a bit ov erwhelming. In following the symmetry-reduction theme in geometric mec hanics from the Euler-Poincar ´ e viewpoint, I ha ve tried to select only the material the studen t will ﬁnd absolutely necessary for solving the problems and exercises, at the lev el of a b e- ginning p ostgraduate studen t. The primary references are [Ma1992], [MaRa1994], [Le2003], [Bl2004], [RaT uSbSoT e2005]. Other v ery useful references are [ArKh1998] and [Ol2000]. The reader ma y see the strong inﬂuences of all these references in these lecture notes, but expressed at a considerably lo wer lev el of mathematical sophistication than the originals. The scope of these lectures is quite limited: a list of the topics in geometric mechanics not included in these lectures would ﬁll volumes! The necessary elements of calculus on smo oth manifolds and the basics of Lie group theory are only brieﬂy described here, b ecause these topics w ere discussed in more depth by other lecturers at the summer sc ho ol. Occasional handouts are included that add a bit more depth in certain k ey topics. The main sub ject of these lecture notes is the use of Lie symmetries in Hamilton’s principle to derive symmetry-reduced equations of motion and to analyze their solutions. The Legendre transformation pro vides the Hamiltonian formulation of these equations in terms of Lie-P oisson brack ets. F or example, we consider Lagrangians in Hamilton’s principle deﬁned on the tangent space T G of a Lie group G . Inv ariance of such a Lagrangian under the action of G leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar ´ e equations. In this case, the inv ariant Lagrangian is deﬁned on the Lie algebra of the group and its Euler-P oincar´ e equations are deﬁned on the dual Lie algebra, where dual is deﬁned by the op eration of taking v ariational deriv ative. On the Hamiltonian side, the Euler-P oincar´ e equations are Lie-Poisson and they possess accompanying momen tum maps, whic h enco de b oth their conserv ation laws and the geometry of their solution space. The standard Euler-Poincar ´ e examples are treated, including particle dynamics, the rigid b ody , the hea vy top and geo desic motion on Lie groups. Additional topics deal with F ermat’s principle, the R 3 P oisson brack et, polarized optical trav eling wa ves, deformable b o dies (Riemann ellipsoids) and shallo w water w av es, including the integrable shallo w water w a ve system kno wn as the Camassa-Holm equation. The lectures end with the semidirect-product Euler-Poincar ´ e reduction theorem for ideal ﬂuid dynamics. This theorem introduces the Euler–P oincar´ e v ariational principle for incompressible and compressible motions of ideal ﬂuids, with applications to geophysical ﬂuids. It also leads to their Lie-P oisson Hamiltonian formulation. Some of these lectures were ﬁrst given at the MASIE (Mec hanics and Symmetry in Europe) summer sc ho ol in 2000 [Ho2005]. I am grateful to the MASIE participants for their helpful remarks and suggestions which led to many improv ements in those lectures. F or their feedbac k and commen ts, I am also grateful to my colleagues at Imp erial College London, esp ecially Colin Cotter, Matthew Dixon, J. D. Gibb on, J. Gibbons, G. Gott wald, J.T. Stuart, J.-L. Thiﬀeault, Cesare T ronci and the studen ts who attended these lectures in my classes at Imperial College. After each class, the students were requested to turn in a response sheet on whic h they answered tw o questions. These questions w ere, “What was this class ab out?” and “What question would y ou like to see pursued in the class?” The D. D. Holm Imp erial College London Applications of P oisson Geometry 6 answ ers to these questions help ed keep the lectures on track with the interests and understanding of the students and it enfranc hised the studen ts because they themselves selected the material in sev eral of the lectures. I am enormously grateful to man y friends and colleagues whose encouragement, advice and supp ort ha ve help ed sustained my interest in this ﬁeld o ver the years. I am particularly grateful to J. E. Marsden, T. S. Ratiu and A. W einstein for their faithful comraderie in many research endea vor s. D. D. Holm Imp erial College London Applications of P oisson Geometry 7 1 In tro duction 1.1 Road map for the course • Spaces – Smo oth Manifolds • Motion – Flo ws φ t ◦ φ s = φ t + s of Lie groups acting on smo oth manifolds • La ws of Motion and discussion of solutions • Newton’s La ws – Newton: dp/dt = F , for momentum p and prescribed force F (on R n historically) – Optimal motion ∗ Euler-Lagrange equations – optimal “action” (Hamilton’s principle) ∗ Geo desic motion – optimal with respect to kinetic energy metric • Lagrangian and Hamiltonian F ormalism – Newton’s La w of motion – Euler-Lagrange theorem – No ether theorem – Euler-P oincar´ e theorem – Kelvin-No ether theorem • Applications and examples – Geo desic motion on a Riemannian manifold – Rigid b ody – geo desic motion on S O (3) – Other geo desic motion, e.g., Riemann ellipsoids on GL (3 , R ) – Hea vy top • Lagrangian mec hanics on Lie groups & Euler-Poincar ´ e (EP) equations – EP( G ), EP equations for geo desics on a Lie group G – EPDiﬀ( R ) for geo desics on Diﬀ( R ) – Pulsons, the singular solutions of EPDiﬀ( R )) wrt an y norm – P eakons, the singular solitons for EPDiﬀ( R , H 1 ), wrt the H 1 norm – EPDiﬀ( R n ) and singular geo desics – Diﬀeons and momen tum maps for EPDiﬀ( R n ) • Euler-P oincar´ e (EP) equations for contin ua – EP semidirect-pro duct reduction theorem – Kelvin-No ether circulation theorem – EP equations with adv ected parameters for geophysical ﬂuid dynamics D. D. Holm Imp erial College London Applications of P oisson Geometry 8 Hamilton ’s principle of stationary action: L agr angians on T R 3 N : Euler-Lagrange equations No ether’s theorem Symmetry = ⇒ cons. la ws Legendre transformation Hamilton’s canonical equations P oisson brack ets Symplectic manifold Momen tum map Reduction b y symmetry G-invariant L agr angians on T G : Euler-P oincar´ e equations Kelvin-No ether theorem Cons. laws are built-in Legendre transformation Lie-Hamilton equations Lie-P oisson brack ets P oisson manifold Momen tum map Reduction to coadjoin t orbits 1.2 Motiv ation for the geometric approach W e b egin with a series of outline sketc hes to motiv ate the geometric approach taken in the course and explain more ab out its con ten t. Wh y is the geometric approac h useful? • Deﬁnes problems on manifolds – co ordinate-free ∗ don’t ha ve to re-do calculations when c hanging co ordinates ∗ more compact ∗ uniﬁed framew ork for expressing ideas and using symmetry • “First principles” approac h – v ariational principles – systematic – uniﬁed approac h e.g. similarity betw een tops and ﬂuid dynamics (semi-direct pro duct), and MHD, and . . . – PO WER Geometric constructions can give useful answ ers without us ha ving to ﬁnd and work with complicated explicit solutions. e.g. stability of rigid b o dy equilibria. Course Outline • Geometrical Structure of Classical Mec hanics – Smo oth manifolds ∗ calculus ∗ tangen t vectors ∗ action principles – Lie groups ∗ ﬂo w prop ert y φ t + s = φ t ◦ φ s ∗ symmetries enco de conserv ation laws in to geometry ∗ ric her than vector spaces – V ariational principles with symmetries ∗ Euler-Lagrange equations → Euler-P oincar´ e equations (more compact) ∗ Tw o main formulations: D. D. Holm Imp erial College London Applications of P oisson Geometry 9 Lagrangian side: Hamilton’s principle No ether’s theorem symmetry = ⇒ cons. la ws momen tum maps Hamiltonian side: Lie-P oisson brack ets cons. laws ⇐ ⇒ symmetries momen tum maps Jacobi iden tity (These t wo views are m utual b eneﬁcial!) • Applications and Mo delling – oscillators & resonance (e.g., LASER) – tops – in tegrable case – ﬂuids – w av es      shallo w water w av es optical pulses solitons Range of topics Rigid bo dy • Euler-Lagrange and Euler-P oincar´ e equations • Kelvin-No ether theorem • Lie-P oisson brack et, Casimirs & coadjoin t orbits • Reconstruction and momen tum maps • The symmetric form of the rigid b ody equations ( ˙ Q = Q Ω, ˙ P = P Ω) • R 3 brac ket and in tersecting lev el surfaces ˙ x = ∇ C × ∇ H = ∇ ( αC + β H ) × ∇ ( γ C + H ) , for α  − β γ = 1 Examples: (1) Con version: rigid b ody ⇐ ⇒ p endulum, (3) F ermat’s principle and ray optics, (2) Self-induced transparency . • Nonlinear oscillators: the n : m resonance • S U (2) rigid b ody , Cayley-Klein parameters and Hopf ﬁbration • The P oincar´ e sphere for p olarization dynamics • 3-w av e resonance, Maxwell-Bloch equations, cavit y resonators, symmetry reduction and the Hopf ﬁbration • 4-w av e resonance, coupled Hopf ﬁbrations, coupled P oincar´ e spheres and coupled rigid b o dies • Higher dimensional rigid b odies – Manak ov in tegrable top on O ( n ) and its sp ectral problem D. D. Holm Imp erial College London Applications of P oisson Geometry 10 • Semi-rigid b odies – geo desic motion on GL (3) and Riemann ellipsoids • Reduction with resp ect to subgroups of GL (3) and Calogero equations Hea vy top • Euler–P oincar´ e v ariational principle for the hea vy top • Kaluza-Klein form ulation of the heavy top Utilit y Kirc hhoﬀ elastica, underw ater vehicles, liquid crystals, stratiﬁed ﬂows, p olarization dynamics of telcom optical pulses General theory • Euler-P oincar´ e semidirect-pro duct reduction theorem • Semidirect-pro duct Lie-P oisson form ulation Shallo w w ater w av es • CH equation – p eak ons (geo desics) • EPDiﬀ equation – (also geo desics) Fluid dynamics • Euler–P oincar´ e v ariational principle for incompressible ideal ﬂuids • Euler–P oincar´ e v ariational principle for compressible ideal ﬂuids Outlo ok: The v ariational principles and the Poisson brack ets for the rigid bo dy and the hea vy top pro vide mo dels of a general construction asso ciated to Euler–Poincar ´ e reduction with resp ect to any Lie group. The Hamiltonian counterpart will b e the semidirect-pro duct Lie-P oisson formulation. W e will often refer to the rigid b o dy and the heavy top for in terpretation and enhanced understanding of the general results. 2 Review Newton, Lagrange & Hamilton • Newton ’s L aw: m ¨ q = F ( q , ˙ q ), inertial frames, uniform motion, etc. • L agr ange’s e quations: d dt ∂ L ∂ ˙ q = ∂ L ∂ q for Lagrangian L ( q , ˙ q , t ). Deﬁned on the tangen t bundle 1 T Q of the conﬁguration space Q with co ordinates ( q , ˙ q ) ∈ T Q , the solution is a curve (or tra jectory) in Q parameterized b y time t . The tangent v ector of the curve q ( t ) through eac h 1 The terms tangent bundle and cotangent bundle are deﬁned in Section 5. F or no w, we may think of the tangen t bundle as the space of positions and velocities. Lik ewise, the cotangent bundle is the space of p ositions and momen ta. D. D. Holm Imp erial College London Applications of P oisson Geometry 11 p oin t q ∈ Q is the v elo cit y ˙ q along the tra jectory that passes though the p oin t q at time t . This v ector is written ˙ q ∈ T q Q . Lagrange’s equations ma y b e expressed compactly in terms of vector ﬁelds and one-forms (diﬀer- en tials). Namely , the Lagrangian vector ﬁeld X L = ˙ q ∂ ∂ q + F ( q , ˙ q ) ∂ ∂ ˙ q acts on the one-form ( ∂ L ∂ ˙ q dq ) just as a time-deriv ative do es, to yield d dt  ∂ L ∂ ˙ q dq  =  d dt ∂ L ∂ ˙ q  dq +  ∂ L ∂ ˙ q  d ˙ q = dL = ⇒ d dt ∂ L ∂ ˙ q = ∂ L ∂ q • Hamiltonian H ( p · q ) = p ˙ q − L and Hamilton ’s c anonic al e quations: ˙ q = ∂ H ∂ p , ˙ p = − ∂ H ∂ q The c onﬁgur ation sp ac e Q has co ordinates q ∈ Q . Its phase space, or cotangen t bundle T ∗ Q has co ordinates ( q , p ) ∈ T ∗ Q . Hamilton’s canonical equations are asso ciated to the c anonic al Poisson br acket for functions on phase space, b y ˙ p = { p , H } , ˙ q = { q , H } ⇐ ⇒ ˙ F ( q , p ) = { F , H } = ∂ F ∂ q ∂ H ∂ p − ∂ F ∂ p ∂ H ∂ q The canonical Poisson brack et has the following familiar properties, whic h may be readily v eri- ﬁed: 1. It is bilinear, 2. sk ew symmetric, { F , H } = − { H , F } , 3. satisﬁes the Leibnitz rule (c hain rule), { F G , H } = { F , H } G + F { G , H } for the pro duct of an y t wo phase space functions F and G , 4. and satisﬁes the Jacobi iden tity { F , { G , H }} + { G , { H , F }} + { H , { F , G }} = 0 for an y three phase space functions F , G and H . Its Leibnitz prop erty (chain rule) prop ert y means the canonical Poisson brac ket is a type of deriv ative. This deriv ation prop ert y of the Poisson brac ket allo ws its use in deﬁning the Hamil- tonian ve ctor ﬁeld X H , b y X H = {· , H } = ∂ H ∂ p ∂ ∂ q − ∂ H ∂ q ∂ ∂ p , for an y phase space function H . The action of X H on phase space functions is giv en by ˙ p = X H p , ˙ q = X H q , and X H ( F G ) = ( X H F ) G + F X H G = ˙ F G + F ˙ G . Th us, solutions of Hamilton’s canonical equations are the characteristic paths of the ﬁrst order linear partial diﬀeren tial op erator X H . That is, X H corresp onds to the time deriv ative along these c haracteristic paths, given b y dt = dq ∂ H /∂ p = dp − ∂ H /∂ q (2.1) The union of these paths in phase space is called the ﬂow of the Hamiltonian v ector ﬁeld X H . D. D. Holm Imp erial College London Applications of P oisson Geometry 12 Prop osition 2.1 (P oisson brac k et as comm utator of Hamiltonian vector ﬁelds) The Pois- son br acket { F , H } is asso ciate d to the c ommutator of the c orr esp onding Hamiltonian ve ctor ﬁelds X F and X H by X { F , H } = X H X F − X F X H =: − [ X F , X H ] Pro of. V eriﬁed by direct computation. Corollary 2.2 Thus, the Jac obi identity for the c anonic al Poisson br acket {· , ·} is asso ciate d to the Jac obi identity for the c ommutator [ · , · ] of the c orr esp onding Hamiltonian ve ctor ﬁelds, [ X F , [ X G , X H ] + [ X G , [ X H , X F ] + [ X H , [ X F , X G ] = 0 . Pro of. This is the Lie algebra prop ert y of Hamiltonian vector ﬁelds, as v eriﬁed b y direct computation. 2.1 Diﬀeren tial forms The diﬀer ential , or exterior derivative of a function F on phase space is written dF = F q dq + F p dp , in which subscripts denote partial deriv atives. F or the Hamiltonian itself, the exterior deriv ativ e and the canonical equations yield dH = H q dq + H p dp = − ˙ pdq + ˙ q dp . The action of a Hamiltonian vector ﬁeld X H on a phase space function F comm utes with its diﬀeren tial, or exterior deriv ative. Thus, d ( X H F ) = X H ( dF ) . This means X H ma y also act as a time deriv ative on diﬀeren tial forms deﬁned on phase space. F or example, it acts on the time-dep endent one-form p dq ( t ) along solutions of Hamilton’s equations as, X H  p dq  = d dt  p dq  = ˙ p dq + p d ˙ q = ˙ p dq − ˙ q dp + d ( p ˙ q ) = − H q dq − H p dp + d ( p ˙ q ) = d ( − H + p ˙ q ) =: dL ( q , p ) up on substituting Hamilton’s canonical equations. The exterior deriv ative of the one-form pdq yields the canonical, or symplectic t wo-form 2 d ( pdq ) = dp ∧ dq 2 The prop erties of diﬀerential forms are summarized in the handouts in sections 3 and 15. D. D. Holm Imp erial College London Applications of P oisson Geometry 13 Here we hav e used the c hain rule for the exterior deriv ative and its prop ert y that d 2 = 0. (The latter amounts to equalit y of cross deriv atives for con tinuous functions.) The result is written in terms of the wedge pro duct ∧ , whic h com bines tw o one-forms (the line elements dq and dp ) in to a tw o-form (the oriented surface element dp ∧ dq = − dq ∧ dp ). As a result, the tw o-form ω = dq ∧ dp represen ting area in phase space is conserved along the Hamiltonian ﬂo ws: X H  dq ∧ dp  = d dt  dq ∧ dp  = 0 This pro ves Theorem 2.3 (Poincar ´ e’s theorem) Hamiltonian ﬂows pr eserve ar e a in phase sp ac e. Deﬁnition 2.4 (Symplectic t w o-form) The phase sp ac e ar e a ω = dq ∧ dp is c al le d the sym- ple ctic two-form. Deﬁnition 2.5 (Symplectic ﬂo ws) Flows that pr eserve ar e a in phase sp ac e ar e said to b e symple ctic. Remark 2.6 (Poincar ´ e’s theorem) Hamiltonian ﬂows ar e symple ctic. D. D. Holm Imp erial College London Applications of P oisson Geometry 14 3 Handout on exterior calculus, symplectic forms and P oincar´ e’s theorem in higher dimensions Exterior calculus on symplectic manifolds is the geometric language of Hamiltonian mec hanics. As an in tro duction and motiv ation for more detailed study , we begin with a preliminary discussion. In diﬀerential geometry , the op eration of c ontr action denoted as in tro duces a pairing b et ween v ector ﬁelds and diﬀerential forms. Con traction is also called substitution of a v ector ﬁeld into a diﬀeren tial form. F or example, there are the dual relations, ∂ q dq = 1 = ∂ p dp , and ∂ q dp = 0 = ∂ p dq A Hamiltonian v ector ﬁeld: X H = ˙ q ∂ ∂ q + ˙ p ∂ ∂ p = H p ∂ q − H q ∂ p = { · , H } satisﬁes X H dq = H p and X H dp = − H q The rule for contraction or substitution of a vector ﬁeld into a diﬀerential form is to sum the substitu- tions of X H o ver the permutations of the factors in the diﬀeren tial form that bring the corresp onding dual basis element in to its leftmost p osition. F or example, substitution of the Hamiltonian vector ﬁeld X H in to the symplectic form ω = dq ∧ dp yields X H ω = X H ( dq ∧ dp ) = ( X H dq ) dp − ( X H dp ) dq In this example, X H dq = H p and X H dp = − H q , so X H ω = H p dp + H q dq = dH whic h follows because ∂ q dq = 1 = ∂ p dp and ∂ q dp = 0 = ∂ p dq . This calculation prov es Theorem 3.1 (Hamiltonian v ector ﬁeld) The Hamiltonian ve ctor ﬁeld X H = { · , H } satisﬁes X H ω = dH with ω = dq ∧ dp (3.1) Relation (3.1) ma y b e tak en as the deﬁnition of a Hamiltonian vector ﬁeld. As a consequence of this formula, the ﬂo w of X H preserv es the closed exact t wo form ω for an y Hamiltonian H . This preserv ation may b e veriﬁed b y a formal calculation using (3.1). Along ( dq /dt, dp/dt ) = ( ˙ q , ˙ p ) = ( H p , − H q ), w e hav e dω dt = d ˙ q ∧ dp + dq ∧ d ˙ p = dH p ∧ dp − dq ∧ dH q = d ( H p dp + H q dq ) = d ( X H ω ) = d ( dH ) = 0 The ﬁrst step uses the c hain rule for diﬀerential forms and the third and last steps use the prop ert y of the exterior deriv ative d that d 2 = 0 for con tinuous forms. The latter is due to equality of cross deriv atives H pq = H q p and an tisymmetry of the wedge product: dq ∧ dp = − dp ∧ dq . Consequen tly , the relation d ( X H ω ) = d 2 H = 0 for Hamiltonian v ector ﬁelds shows D. D. Holm Imp erial College London Applications of P oisson Geometry 15 Theorem 3.2 (Poincar ´ e’s theorem for one degree of freedom) The ﬂow of a Hamiltonian ve ctor ﬁeld is symple ctic , which me ans it pr eserves the phase-sp ac e ar e a, or two-form, ω = dq ∧ dp . Deﬁnition 3.3 (Cartan’s form ula for the Lie deriv ative) The op er ation of Lie derivative of a diﬀer ential form ω by a ve ctor ﬁeld X H is deﬁne d by £ X H ω := d ( X H ω ) + X H dω (3.2) Corollary 3.4 Be c ause dω = 0 , the symple ctic pr op erty dω /dt = d ( X H ω ) = 0 in Poinc ar´ e’s The or em 3.2 may b e r ewritten using Lie derivative notation as 0 = dω dt = £ X H ω := d ( X H ω ) + X H dω =: (div X H ) ω . (3.3) The last e quality deﬁnes the diver genc e of the ve ctor ﬁeld X H in terms of the Lie derivative. Remark 3.5 • R elation (3.3) asso ciates Hamiltonian dynamics with the symple ctic ﬂow in phase sp ac e of the Hamiltonian ve ctor ﬁeld X H , which is diver genc eless with r esp e ct to the symple ctic form ω . • The Lie derivative op er ation deﬁne d in (3.3) is e quivalent to the time derivative along the char ac- teristic p aths (ﬂow) of the ﬁrst or der line ar p artial diﬀer ential op er ator X H , which ar e obtaine d fr om its char acteristic e quations in (2.1). This is the dynamic al me aning of the Lie derivative £ X H in (3.2) for which invarianc e £ X H ω = 0 gives the ge ometric deﬁnition of symple ctic ﬂows in phase sp ac e. Theorem 3.6 (Poincar ´ e’s theorem for N degrees of freedom) F or a system of N p articles, or N de gr e es of fr e e dom, the ﬂow of a Hamiltonian ve ctor ﬁeld pr eserves e ach subvolume in the phase sp ac e T ∗ R N . That is, let ω n ≡ dq n ∧ dp n b e the symple ctic form expr esse d in terms of the p osition and momentum of the n − th p article. Then d ω M dt = 0 , for ω M = Π M n =1 ω n , ∀ M ≤ N . The pro of of the preserv ation of these Poinc ar´ e invariants ω M with M = 1 , 2 , . . . , N follo ws the same pattern as the v eriﬁcation abov e for a single degree of freedom. Basically , this is because each factor ω n = dq n ∧ dp n in the wedge product of symplectic forms is preserv ed by its corresp onding Hamiltonian ﬂo w in the sum X H = M X n =1  ˙ q n ∂ ∂ q n + ˙ p n ∂ ∂ p n  = M X n =1  H p n ∂ q n − H q n ∂ p n  = M X n =1 X H n = { · , H } That is, £ X H n ω M v anishes for each term in the sum £ X H ω M = P M n =1 £ X H n ω M since ∂ q m dq n = δ mn = ∂ p m dp n and ∂ q m dp n = 0 = ∂ p m dq n . D. D. Holm Imp erial College London Applications of P oisson Geometry 16 4 F ermat’s theorem in geometrical ra y optics 4.1 F ermat’s principle: Rays tak e paths of least optical length In geometrical optics, the ra y path is determined by F ermat’s principle of least optical length, δ Z n ( x, y , z ) ds = 0 . Here n ( x, y , z ) is the index of refraction at the spatial p oin t ( x, y , z ) and ds is the elemen t of arc length along the ra y path through that p oin t. Cho osing coordinates so that the z − axis coincides with the optical axis (the general direction of propagation), giv es ds = [( dx ) 2 + ( dy ) 2 + ( dz ) 2 ] 1 / 2 = [1 + ˙ x 2 + ˙ y 2 ] 1 / 2 dz , with ˙ x = dx/dz and ˙ y = dy /dz . Thus, F ermat’s principle can b e written in Lagrangian form, with z pla ying the role of time, δ Z L ( x, y , ˙ x, ˙ y , z ) dz = 0 . Here, the optical Lagrangian is, L ( x, y , ˙ x, ˙ y , z ) = n ( x, y , z )[1 + ˙ x 2 + ˙ y 2 ] 1 / 2 =: n/γ , or, equiv alently , in tw o-dimensional v ector notation with q = ( x, y ), L ( q , ˙ q , z ) = n ( q , z )[1 + | ˙ q | 2 ] 1 / 2 =: n/γ with γ = [1 + | ˙ q | 2 ] − 1 / 2 ≤ 1 . Consequen tly , the vector Euler-Lagrange equation of the ligh t rays is d ds  n d q ds  = γ d dz  nγ d q dz  = ∂ n ∂ q . The momen tum p canonically conjugate to the ra y path p osition q in an “image plane”, or on an “image screen”, at a ﬁxed v alue of z is giv en b y p = ∂ L ∂ ˙ q = nγ ˙ q whic h satisﬁes | p | 2 = n 2 (1 − γ 2 ) . This implies the v elo cit y ˙ q = p / ( n 2 − | p | 2 ) 1 / 2 . Hence, the momentum is real-v alued and the Lagrangian is hyperregular, provided n 2 − | p | 2 > 0. When n 2 = | p | 2 , the ra y tra jectory is v ertical and has gr azing incidenc e with the image screen. Deﬁning sin θ = dz /ds = γ leads to | p | = n cos θ , and giv es the follo wing geometrical picture of the ray path. Along the optical axis (the z − axis) each image plane normal to the axis is pierced at a point q = ( x, y ) by a vector of magnitude n ( q , z ) tangen t to the ray path. This v ector mak es an angle θ to the plane. The pro jection of this vector on to the image plane is the canonical momentum p . This picture of the ray paths captures all but the rays of grazing incidence to the image planes. Suc h grazing rays are ignored in what follo ws. P assing no w via the usual Legendre transformation from the Lagrangian to the Hamiltonian de- scription giv es H = p · ˙ q − L = nγ | ˙ q | 2 − n/γ = − nγ = −  n ( q , z ) 2 − | p | 2  1 / 2 D. D. Holm Imp erial College London Applications of P oisson Geometry 17 Th us, in the geometrical picture, the component of the tangen t v ector of the ray-path along the optical axis is (min us) the Hamiltonian, i.e. n ( q , z ) sin θ = − H . The phase space description of the ra y path now follo ws from Hamilton’s equations, ˙ q = ∂ H ∂ p = − 1 H p , ˙ p = − ∂ H ∂ q = − 1 2 H ∂ n 2 ∂ q . Remark 4.1 (T ranslation in v arian t media) If n = n ( q ) , so that the me dium is tr anslation in- variant along the optic al axis, z , then H = − n sin θ is c onserve d. (Conservation of H at an interfac e is Snel l’s law.) F or tr anslation-invariant me dia, the ve ctor r ay-p ath e quation simpliﬁes to ¨ q = − 1 2 H 2 ∂ n 2 ∂ q , Newtonian dynamics for q ∈ R 2 . Thus, in this c ase ge ometric al r ay tr acing r e duc es to “Newtonian dynamics” in z , with p otential − n 2 ( q ) and with “time” r esc ale d along e ach p ath by the value of √ 2 H determine d fr om the initial c onditions for e ach r ay. 4.2 Axisymmetric, translation in v arian t materials In axisymmetric, translation inv arian t media, the index of refraction is a function of the radius alone. Axisymmetry implies an additional constan t of motion and, hence, reduction of the Hamiltonian system for the ligh t rays to phase plane analysis. F or such media, the index of refraction satisﬁes n ( q , z ) = n ( r ) , r = | q | . P assing to p olar co ordinates ( r , φ ) with q = ( x, y ) = r (cos φ, sin φ ) leads in the usual w ay to | p | 2 = p 2 r + p 2 φ /r 2 . Consequen tly , the optical Hamiltonian, H = −  n ( r ) 2 − p 2 r − p 2 φ /r 2  1 / 2 is indep enden t of the azimuthal angle φ ; so its canonically conjugate “angular momentum” p φ is conserv ed. Using the relation q · p = r p r leads to an interpretation of p φ in terms of the image-screen phase space v ariables p and q . Namely , | p × q | 2 = | p | 2 | q | 2 − ( p · q ) 2 = p 2 φ The conserv ed quan tity p φ = p × q = y p x − xp y is called the sk ewness function, or the P etzv al in v ariant for axisymmetric media. V anishing of p φ o ccurs for meridional r ays , for which p and q are collinear in the image plane. On the other hand, p φ tak es its maximum v alue for sagittal r ays , for whic h p · q = 0, so that p and q are orthogonal in the image plane. Exercise 4.2 (Axisymmetric, translation in v ariant materials) Write Hamilton ’s c anonic al e qua- tions for axisymmetric, tr anslation invariant me dia. Solve these e quations for the c ase of an optic al ﬁb er with r adial ly gr ade d index of r efr action in the fol lowing form: n 2 ( r ) = λ 2 + ( µ − ν r 2 ) 2 , λ, µ, ν = constants , by r e ducing the pr oblem to phase plane analysis. How do es the phase sp ac e p ortr ait diﬀer b etwe en p φ = 0 and p φ 6 = 0 ? Show that for p φ 6 = 0 the pr oblem r e duc es to a Duﬃng oscil lator in a r otating fr ame, up to a r esc aling of time by the value of the Hamiltonian on e ach r ay “orbit.” D. D. Holm Imp erial College London Applications of P oisson Geometry 18 4.3 The Petzv al inv arian t and its Poisson brac k et relations The sk ewness function S = p φ = p × q = y p x − xp y generates rotations of phase space, of q and p join tly , each in its plane, around the optical axis. Its square, S 2 (called the P etzv al in v arian t) is conserved for ray optics in axisymmetric media. That is, { S 2 , H } = 0 for optical Hamiltonians of the form, H = −  n ( | q | 2 ) 2 − | p | 2  1 / 2 . W e deﬁne the axisymmetric inv ariant coordinates by the map T ∗ R 2 7→ R 3 ( q , p ) 7→ ( X, Y , Z ), X = | q | 2 ≥ 0 , Y = | p | 2 ≥ 0 , Z = p · q . The follo wing Poisson brac k et relations hold { S 2 , X } = 0 , { S 2 , Y } = 0 , { S 2 , Z } = 0 , since rotations preserv e dot pro ducts. In terms of these inv ariant co ordinates, the P etzv al inv arian t and optical Hamiltonian satisfy S 2 = X Y − Z 2 ≥ 0 , and H 2 = n 2 ( X ) − Y ≥ 0 . The lev el sets of S 2 are h yp erboloids of revolution around the X = Y axis, extending up through the in terior of the S = 0 cone, and lying b et ween the X − and Y − axes. The lev el sets of H 2 dep end on the functional form of the index of refraction, but they are Z − indep enden t. 4.4 R 3 P oisson brack et for ra y optics The P oisson brack ets among the axisymmetric v ariables X , Y and Z close among themselv es, { X , Y } = 4 Z , { Y , Z } = − 2 Y , { Z , X } = − 2 X . These P oisson brack ets deriv e from a single R 3 P oisson brack et for X = ( X, Y , Z ) given b y { F , H } = −∇ S 2 · ∇ F × ∇ H Consequen tly , w e may re-express the equations of Hamiltonian ray optics in axisymmetric media with H = H ( X , Y ) as ˙ X = ∇ S 2 × ∇ H . with Casimir S 2 , for which { S 2 , H } = 0, for ev ery H . Th us, the ﬂow preserves volume (div ˙ X = 0) and the evolution tak es place on in tersections of lev el surfaces of the axisymmetric media in v ariants S 2 and H ( X , Y ). 4.5 Recognition of the Lie-P oisson brac k et for ra y optics The Casimir inv ariant S 2 = X Y − Z 2 is quadratic. In such cases, one ma y write the R 3 P oisson brac ket in the suggestiv e form, { F , H } = − C k ij X k ∂ F ∂ X i ∂ H ∂ X j D. D. Holm Imp erial College London Applications of P oisson Geometry 19 In this particular case, C 3 12 = 4, C 2 23 = 2 and C 1 31 = 2 and the rest either v anish, or are obtained from an tisymmetry of C k ij under exc hange of an y pair of its indices. These v alues are the structure constan ts of an y of the Lie algebras sp (2 , R ), so (2 , 1), su (1 , 1), or sl (2 , R ). Th us, the reduced description of Hamiltonian ra y optics in terms of axisymmetric R 3 v ariables is said to b e “Lie-Poisson” on the dual space of any of these Lie algebras, say , sp (2 , R ) ∗ for deﬁniteness. W e will hav e more to say ab out Lie-P oisson brack ets later, when w e reach the Euler-Poincar ´ e reduction theorem. Exercise 4.3 Consider the R 3 Poisson br acket { f , h } = − ∇ c · ∇ f × ∇ h (4.1) L et c = x T · C x b e a quadr atic form on R 3 , and let C b e the asso ciate d symmetric 3 × 3 matrix. Show that this is the Lie-Poisson br acket for the Lie algebr a structur e [ u , v ] C = C ( u × v ) What is the underlying matrix Lie algebr a? What ar e the c o adjoint orbits of this Lie algebr a? Remark 4.4 (Coadjoint orbits) As one might exp e ct, the c o adjoint orbits of the gr oup S P (2 , R ) ar e the hyp erb oloids c orr esp onding to the level sets of S 2 . Remark 4.5 As we shal l se e later, the map T ∗ R 2 7→ sp (2 , R ) ∗ taking ( q , p ) 7→ ( X , Y , Z ) is an example of a momentum map . 5 Geometrical Structure of Classical Mechanics 5.1 Manifolds Conﬁguration space: co ordinates q ∈ M , where M is a smo oth manifold. φ β ◦ φ − 1 α is a smo oth c hange of v ariables. F or later, smo oth coordinate transformations: q → Q with dQ = ∂ Q ∂ q dq Deﬁnition 5.1 A smo oth manifold M is a set of p oints to gether with a ﬁnite (or p erhaps c ountable) set of subsets U α ⊂ M and 1-to-1 mappings φ α : U α → R n such that 1. S α U α = M 2. F or every nonempty interse ction U α ∩ U β , the set φ α ( U α ∩ U β ) is an op en subset of R n and the 1-to-1 mapping φ β ◦ φ − 1 α is a smo oth function on φ α ( U α ∩ U β ) . Remark 5.2 The sets U α in the deﬁnition ar e c al le d c o or dinate charts . The mappings φ α ar e c al le d c o or dinate functions or lo c al c o or dinates . A c ol le ction of charts satisfying 1 and 2 is c al le d an atlas . Condition 3 al lows the deﬁnition of manifold to b e made indep endently of a choic e of atlas. A set of charts satisfying 1 and 2 c an always b e extende d to a maximal set; so, in pr actic e, c onditions 1 and 2 deﬁne the manifold. Example 5.3 Manifolds often arise as interse ctions of zer o level sets M =  x   f i ( x ) = 0 , i = 1 , . . . , k  , for a given set of functions f i : R n → R , i = 1 , . . . , k . If the gr adients ∇ f i ar e line arly indep endent, or mor e gener al ly if the r ank of {∇ f ( x ) } is a c onstant r for al l x, then M is a smo oth manifold of dimension n − r . The pr o of uses the Implicit F unction The or em to show that an ( n − r ) − dimensional c o or dinate chart may b e deﬁne d in a neighb orho o d of e ach p oint on M . In this situation, the set M is c al le d a submanifold of R n (se e [L e2003]). D. D. Holm Imp erial College London Applications of P oisson Geometry 20 Deﬁnition 5.4 If r = k , then the map { f i } is c al le d a submersion . Exercise 5.5 Pr ove that al l submersions ar e submanifolds (se e [L e2003]). Deﬁnition 5.6 (T angen t space to level sets) L et M =  x   f i ( x ) = 0 , i = 1 , . . . , k  b e a manifold in R n . The tangent sp ac e at e ach x ∈ M , is deﬁne d by T x M =  v ∈ R n   ∂ f i ∂ x a ( x ) v a = 0 , i = 1 , . . . , k  . Note: we use the summation c onvention , i.e. r ep e ate d indic es ar e summe d over their r ange. Remark 5.7 The tangent sp ac e is a line ar ve ctor sp ac e. Example 5.8 (T angen t space to the sphere in R 3 ) Example 5.9 (T angen t space to the sphere in R 3 ) The spher e S 2 is the set of p oints ( x, y , z ) ∈ R 3 solving x 2 + y 2 + z 2 = 1 . The tangent sp ac e to the spher e at such a p oint ( x, y , z ) is the plane c ontaining ve ctors ( u, v , w ) satisfying xu + y v + z w = 0 . Deﬁnition 5.10 [T angent bund le] The tangent bund le of a manifold M , denote d by T M , is the smo oth manifold whose underlying set is the disjoint union of the tangent sp ac es to M at the p oints x ∈ M ; that is, T M = [ x ∈ M T x M Thus, a p oint of T M is a ve ctor v which is tangent to M at some p oint x ∈ M . Example 5.11 (T angen t bundle T S 2 of S 2 ) The tangent bund le T S 2 of S 2 ∈ R 3 is the union of the tangent sp ac es of S 2 : T S 2 =  ( x, y , z ; u, v , w ) ∈ R 6   x 2 + y 2 + z 2 = 1 and xu + yv + z w = 0  . Remark 5.12 (Dimension of tangen t bundle T S 2 ) Deﬁning T S 2 r e quir es two indep endent c on- ditions in R 6 ; so dim T S 2 = 4 . Exercise 5.13 Deﬁne the spher e S n − 1 in R n . What is the dimension of its tangent sp ac e T S n − 1 ? Example 5.14 (The t w o stereographic pro jections of S 2 → R 2 ) The unit spher e S 2 = { ( x, y , z ) : x 2 + y 2 + z 2 = 1 } is a smo oth two-dimensional manifold r e alize d as a submersion in R 3 . L et U N = S 2 \{ 0 , 0 , 1 } , and U S = S 2 \{ 0 , 0 , − 1 } b e the subsets obtaine d by deleting the North and South p oles of S 2 , r esp e ctively. L et χ N : U N → ( ξ N , η N ) ∈ R 2 , and χ S : U S → ( ξ S , η S ) ∈ R 2 D. D. Holm Imp erial College London Applications of P oisson Geometry 21 b e ster e o gr aphic pr oje ctions fr om the North and South p oles onto the e quatorial plane, z = 0 . Thus, one may plac e two diﬀer ent c o or dinate p atches in S 2 interse cting everywher e exc ept at the p oints along the z − axis at z = 1 (North p ole) and z = − 1 (South p ole). In the e quatorial plane z = 0 , one may deﬁne two sets of (right-hande d) c o or dinates, φ α : U α → R 2 \{ 0 } , α = N , S , obtaine d by the fol lowing two ster e o gr aphic pr oje ctions fr om the North and South p oles: (1) (valid everywher e exc ept z = 1 ) φ N ( x, y , z ) = ( ξ N , η N ) =  x 1 − z , y 1 − z  , (2) (valid everywher e exc ept z = − 1 ) φ S ( x, y , z ) = ( ξ S , η S ) =  x 1 + z , − y 1 + z  . (The two c omplex planes ar e identiﬁe d diﬀer ently with the plane z = 0 . A n orientation-r eversal is ne c essary to maintain c onsistent c o or dinates on the spher e.) One may che ck dir e ctly that on the overlap U N ∩ U S the map, φ N ◦ φ − 1 S : R 2 \{ 0 } → R 2 \{ 0 } is a smo oth diﬀe omorphism, given by the inversion φ N ◦ φ − 1 S ( x, y ) =  x x 2 + y 2 , y x 2 + y 2  . Exercise 5.15 Construct the mapping fr om ( ξ N , η N ) → ( ξ S , η S ) and verify that it is a diﬀe omorphism in R 2 \{ 0 } . Hint: (1 + z )(1 − z ) = 1 − z 2 = x 2 + y 2 . Answ er 5.16 ( ξ S , − η S ) = 1 − z 1 + z ( ξ N , η N ) = 1 ξ 2 N + η 2 N ( ξ N , η N ) . The map ( ξ N , η N ) → ( ξ S , η S ) is smo oth and invertible exc ept at ( ξ N , η N ) = (0 , 0) . Example 5.17 If we start with two identic al cir cles in the xz -plane, of r adius r and c enter e d at x = ± 2 r , then r otate them r ound the z axis in R 3 , we get a torus, written T 2 . It’s a manifold. Exercise 5.18 If we b e gin with a ﬁgur e eight in the xz -plane, along the x axis and c enter e d at the origin, and spin it r ound the z axis in R 3 , we get a “pinche d surfac e” that lo oks like a spher e that has b e en “pinche d” so that the north and south p oles touch. Is this a manifold? Pr ove it. Answ er 5.19 The origin has a neighb ourho o d diﬀe omorphic to a double c one. This is not diﬀe omor- phic to R 2 . A pr o of of this is that, if the origin of the c one is r emove d, two c omp onents r emain; while if the origin of R 2 is r emove d, only one c omp onent r emains. Remark 5.20 The spher e wil l app e ar in sever al examples as a r e duc e d sp ac e in which motion takes plac e after applying a symmetry. R e duction by symmetry is asso ciate d with a classic al topic in c elestial me chanics known as normal form the ory. R e duction may b e “singular,” in which c ase it le ads to “p ointe d” sp ac es that ar e smo oth manifolds exc ept at one or mor e p oints. F or example diﬀer ent r esonanc es of c ouple d oscil lators c orr esp ond to the fol lowing r e duc e d sp ac es: 1:1 r esonanc e – spher e; 1:2 r esonanc e – pinche d spher e with one c one p oint; 1:3 r esonanc e – pinche d spher e with one cusp p oint; 2:3 r esonanc e – pinche d spher e with one c one p oint and one cusp p oint. D. D. Holm Imp erial College London Applications of P oisson Geometry 22 5.2 Motion: T angen t V ectors and Flo ws En visioning our later considerations of dynamical systems, we shall consider motion along curv es c ( t ) parameterised by time t on a smo oth manifold M . Supp ose these curves are tra jectories of a ﬂow φ t of a v ector ﬁeld. W e anticipate this means φ t ( c (0)) = c ( t ) and φ t ◦ φ s = φ t + s (ﬂo w prop ert y). The ﬂo w will b e tangen t to M along the curve. T o deal with suc h ﬂows, we will need the concept of tangent ve ctors . Recall from Deﬁnition 5.10 that the tangent bundle of M is T M = [ x ∈ M T x M . W e will now add a bit more to that deﬁnition. The tangent bundle is an example of a more general structure than a manifold. Deﬁnition 5.21 (Bundle) A bund le c onsists of a manifold B , another manifold M c al le d the “b ase sp ac e” and a pr oje ction b etwe en them Π : B → M . L o c al ly, in smal l enough r e gions of x the inverse images of the pr oje ction Π exist. These ar e c al le d the ﬁb ers of the bund le. Thus, subsets of the bund le B lo c al ly have the structur e of a Cartesian pr o duct. A n example is ( B , M , Π) c onsisting of ( R 2 , R 1 , Π : R 2 → R 1 ) . In this c ase, Π : ( x, y ) ∈ R 2 → x ∈ R 1 . Likewise, the tangent bund le c onsists of M , T M and a map τ M : T M → M . Let x =  x 1 , . . . , x n  b e local co ordinates on M , and let v =  v 1 , . . . , v n  b e components of a tangen t vector. T x M =  v ∈ R n   ∂ f i ∂ x · v = 0 , i = 1 , . . . , m  for M =  x ∈ R n   f i ( x ) = 0 , i = 1 , . . . , m  These 2 n n umbers ( x, v ) give lo cal co ordinates on T M , whose dimension is dim T M = 2 dim M . The tangent bund le pr oje ction is a map τ M : T M → M which tak es a tangent vector v to a p oin t x ∈ M where the tangen t v ector v is attac hed (that is, v ∈ T x M ). The inv erse of this pro jection τ − 1 M ( x ) is called the ﬁb er ov er x in the tangent bundle. V ector ﬁelds, in tegral curv es and ﬂo ws Deﬁnition 5.22 A ve ctor ﬁeld on a manifold M is a map X : M → T M that assigns a ve ctor X ( x ) at e ach p oint x ∈ M . This implies that τ M ◦ X = I d. Deﬁnition 5.23 An inte gr al curve of X with initial c onditions x 0 at t = 0 is a diﬀer entiable map c :] a, b [ → M , wher e ] a, b [ is an op en interval c ontaining 0 , such that c (0) = 0 and c 0 ( t ) = X ( c ( t )) for al l t ∈ ] a, b [ . Remark 5.24 A standar d r esult fr om the the ory of or dinary diﬀer ential e quations states that X b eing Lipschitz implies its inte gr al curves ar e unique and C 1 [CoL e1984]. The inte gr al curves c ( t ) ar e diﬀer entiable for smo oth X . D. D. Holm Imp erial College London Applications of P oisson Geometry 23 5.3 Summary Deﬁnition 5.25 The ﬂow of X is the c ol le ction of maps φ t : M → M , wher e t → φ t ( x ) is the inte gr al curve of X with initial c ondition x . Remark 5.26 1. Existenc e and uniqueness r esults for solutions of c 0 ( t ) = X ( c ( t )) guar ante e that ﬂow φ of X is smo oth in ( x, t ) , for smo oth X . 2. Uniqueness implies the ﬂow pr op erty φ t + s = φ t ◦ φ s , ( F P ) for initial c ondition φ 0 = I d. 3. The ﬂow pr op erty (FP) gener alizes to the nonline ar c ase the familiar line ar situation wher e M is a ve ctor sp ac e, X ( x ) = Ax is a line ar ve ctor ﬁeld for a b ounde d line ar op er ator A , and φ t ( x ) = e At x . Diﬀeren tials of functions and the cotangen t bundle W e are now ready to deﬁne diﬀerentials of smooth functions and the cotangent bundle. Let f : M → R be a smo oth function. W e diﬀerentiate f at x ∈ M to obtain T x f : T x M → T f ( x ) R . As is standard, w e iden tify T f ( x ) R with R itself, thereby obtaining a linear map d f ( x ) : T x M → R . The result d f ( x ) is an element of the cotangen t space T ∗ x M , the dual space of the tangen t space T x M . The natural pairing b et ween elements of the tangent space and the cotangent space is denoted as h· , ·i : T ∗ x M × T x M 7→ R . In co ordinates, the linear map d f ( x ) : T x M → R ma y b e written as the directional deriv ative, h d f ( x ) , v i = d f ( x ) · v = ∂ f ∂ x i · v i , for all v ∈ T x M . (Reminder: the summation con ven tion is in tended o ver rep eated indices.) Hence, elemen ts d f ( x ) ∈ T ∗ x M are dual to vectors v ∈ T x M with resp ect to the pairing h· , ·i . Deﬁnition 5.27 d f is the diﬀer ential of the function f . Deﬁnition 5.28 The dual sp ac e of the tangent bund le T M is the c otangent bund le T ∗ M . That is, ( T x M ) ∗ = T ∗ x M and T ∗ M = [ x T ∗ x M . Th us, replacing v ∈ T x M with d f ∈ T ∗ x M , for all x ∈ M and for all smo oth functions f : M → R , yields the c otangent bund le T ∗ M . D. D. Holm Imp erial College London Applications of P oisson Geometry 24 Diﬀeren tial bases When the basis of vector ﬁelds is denoted as ∂ ∂ x i for i = 1 , . . . , n , its dual basis is often denoted as dx i . In this notation, the diﬀerential of a function at a p oin t x ∈ M is expressed as d f ( x ) = ∂ f ∂ x i dx i The corresp onding pairing h· , ·i of bases is written in this notation as  dx j , ∂ ∂ x i  = δ j i Here δ j i is the Kroneck er delta, whic h equals unit y for i = j and v anishes otherwise. That is, deﬁning T ∗ M requires a pairing h· , ·i : T ∗ M × T M → R . (Diﬀeren t pairings exist for curvilinear co ordinates, Riemannian manifolds, etc.) 6 Deriv ativ es of diﬀeren tiable maps – the tangen t lift W e next deﬁne deriv atives of diﬀeren tiable maps b et w een manifolds (tangent lifts). W e expect that a smo oth map f : U → V from a c hart U ⊂ M to a chart V ⊂ N , will lift to a map b et ween the tangent bundles T M and T N so as to make sense from the viewp oint of ordinary calculus, U × R m ⊂ T M − → V × R n ⊂ T N  q 1 , . . . , q m ; X 1 , . . . , X m  7− →  Q 1 , . . . , Q n ; Y 1 , . . . , Y n  Namely , the relations betw een the v ector ﬁeld comp onen ts should b e obtained from the diﬀeren tial of the map f : U → V . Perhaps not unexp ectedly , these vector ﬁeld comp onents will b e related b y Y i ∂ ∂ Q i = X j ∂ ∂ q j , so Y i = ∂ Q i ∂ q j X j , in whic h the quantit y called the tangent lift T f = ∂ Q ∂ q of the function f arises from the chain rule and is equal to the Jacobian for the transformation T f : T M 7→ T N . The dual of the tangent lift is the cotangent lift, explained later in section 10.1. Roughly sp eaking, the c otangent lift of the function f , T ∗ f = ∂ q ∂ Q arises from β i dQ i = α j dq j , so β i = α j ∂ q j ∂ Q i and T ∗ f : T ∗ N 7→ T ∗ M . Note the directions of these maps: T f : q , X ∈ T M 7→ Q, Y ∈ T N f : q ∈ M 7→ Q ∈ N T ∗ f : Q , β ∈ T ∗ N 7→ q , α ∈ T ∗ M (map go es the other w a y , see the picture ) D. D. Holm Imp erial College London Applications of P oisson Geometry 25 6.1 Summary remarks ab out deriv ativ es on manifolds Deﬁnition 6.1 (Diﬀerentiable map) A map f : M → N fr om manifold M to manifold N is said to b e diﬀer entiable (r esp. C k ) if it is r epr esente d in lo c al c o or dinates on M and N by diﬀer entiable (r esp. C k ) functions. Deﬁnition 6.2 (Deriv ative of a diﬀerentiable map) The derivative of a diﬀer entiable map f : M → N at a p oint x ∈ M is deﬁne d to b e the line ar map T x f : T x M → T x N c onstructe d, as fol lows. F or v ∈ T x M , cho ose a curve c ( t ) that maps an op en interval t ∈ ( − ,  ) ar ound the p oint t = 0 to the manifold M c : ( − ,  ) − → M with c (0) = x and velo city ve ctor c 0 (0) := dc dt    t =0 = v . Then T x f · v is the velo city ve ctor at t = 0 of the curve f ◦ c : R → N . That is, T x f · v = d dt f ( c ( t ))    t =0 = ∂ f ∂ c d dt c ( t )    t =0 Deﬁnition 6.3 The union T f = S x T x f of the derivatives T x f : T x M → T x N over p oints x ∈ M is c al le d the tangent lift of the map f : M → N . Remark 6.4 The chain-rule deﬁnition of the derivative T x f of a diﬀer entiable map at a p oint x dep ends on the function f and the ve ctor v . Other de gr e es of diﬀer entiability ar e p ossible. F or example, if M and N ar e manifolds and f : M → N is of class C k +1 , then the tangent lift (Jac obian) T x f : T x M → T x N is C k . Exercise 6.5 L et φ t : S 2 → S 2 r otate p oints on S 2 ab out a ﬁxe d axis thr ough an angle ψ ( t ) . Show that φ t is the ﬂow of a c ertain ve ctor ﬁeld on S 2 . Exercise 6.6 L et f : S 2 → R b e deﬁne d by f ( x, y , z ) = z . Compute d f using spheric al c o or dinates ( θ , φ ) . Exercise 6.7 Compute the tangent lifts for the two ster e o gr aphic pr oje ctions of S 2 → R 2 in example 5.14. That is, assuming ( x, y , z ) dep end smo othly on t , ﬁnd 1. How ( ˙ ξ N , ˙ η N ) dep end on ( ˙ x, ˙ y , ˙ z ) . Likewise, for ( ˙ ξ S , ˙ η S ) . 2. How ( ˙ ξ N , ˙ η N ) dep end on ( ˙ ξ S , ˙ η S ) . Hint: R e c al l (1 + z )(1 − z ) = 1 − z 2 = x 2 + y 2 and use x ˙ x + y ˙ y + z ˙ z = 0 when ( ˙ x, ˙ y , ˙ z ) is tangent to S 2 at ( x, y , z ) . D. D. Holm Imp erial College London Applications of P oisson Geometry 26 7 Lie groups and Lie algebras 7.1 Matrix Lie groups Deﬁnition 7.1 A gr oup is a set of elements with 1. A binary pr o duct (multiplic ation), G × G → G The pr o duct of g and h is written g h The pr o duct is asso ciative, ( g h ) k = g ( hk ) 2. Identity element e : eg = g and g e = g , ∀ g ∈ G 3. Inverse op er ation G → G , so that g g − 1 = g − 1 g = e Deﬁnition 7.2 A Lie gr oup is a smo oth manifold G which is a gr oup and for which the gr oup op er ations of multiplic ation, ( g , h ) → g h for g , h ∈ G , and inversion, g → g − 1 with g g − 1 = g − 1 g = e , ar e smo oth. Deﬁnition 7.3 A matrix Lie gr oup is a set of invertible n × n matric es which is close d under matrix multiplic ation and which is a submanifold of R n × n . The c onditions showing that a matrix Lie gr oup is a Lie gr oup ar e e asily che cke d: A matrix Lie gr oup is a manifold, b e c ause it is a submanifold of R n × n Its gr oup op er ations ar e smo oth, sinc e they ar e algebr aic op er ations on the matrix entries. Example 7.4 (The general linear group GL ( n, R ) ) The matrix Lie gr oup GL ( n, R ) is the gr oup of line ar isomorphisms of R n to itself. The dimension of the matric es in GL ( n, R ) is n 2 . Prop osition 7.5 L et K ∈ GL ( n, R ) b e a symmetric matrix, K T = K . Then the sub gr oup S of GL ( n, R ) deﬁne d by the mapping S = { U ∈ GL ( n, R ) | U T K U = K } is a submanifold of R n × n of dimension n ( n − 1) / 2 . Remark 7.6 The sub gr oup S le aves invariant a c ertain symmetric quadr atic form under line ar tr ans- formations, S × R n → R n given by x → U x , sinc e x T K x = x T U T K U x . So the matric es U ∈ S change the b asis for this quadr atic form, but they le ave its value unchange d. Thus, S is the isotr opy sub gr oup of the quadr atic form asso ciate d with K . Pro of. Is S a subgroup? W e chec k the follo wing three deﬁning prop erties 1. Iden tity: I ∈ S b ecause I T K I = K 2. In verse: U ∈ S = ⇒ U − 1 ∈ S , b ecause K = U − T ( U T K U ) U − 1 = U − T ( K ) U − 1 D. D. Holm Imp erial College London Applications of P oisson Geometry 27 3. Closed under m ultiplication: U, V ∈ S = ⇒ U V ∈ S , b ecause ( U V ) T K U V = V T ( U T K U ) V = V T ( K ) V = K Hence, S is a subgroup of GL ( n, R ). Is S is a submanifold of R n × n of dimension n ( n − 1) / 2? Indeed, S is the zero lo cus of the mapping U K U T − K . This mak es it a submanifold, because it turns out to b e a submersion. F or a submersion, the dimension of the lev el set is the dimension of the domain minus the dimension of the range space. In this case, this dimension is n 2 − n ( n + 1) / 2 = n ( n − 1) / 2 Exercise 7.7 Explain why one c an c onclude that the zer o lo cus map for S is a submersion. In p articular, p ay close attention to establishing the c onstant r ank c ondition for the line arization of this map. Solution Here is why S is a submanifold of R n × n . First, S is the zero locus of the mapping U → U T K U − K , (locus map) Let U ∈ S , and let δ U b e an arbitrary elemen t of R n × n . Then linearize to ﬁnd ( U + δ U ) T K ( U + δ U ) − K = U T K U − K + δ U T K U + U T K δ U + O ( δ U ) 2 , . W e ma y conclude that S is a submanifold of R n × n if we can sho w that the linearization of the lo cus map, namely the linear mapping deﬁned b y L ≡ δ U → δ U T K U + U T K δ U , R n × n → R n × n has constan t rank for all U ∈ S . Lemma 7.8 The line arization map L is onto the sp ac e of n × n of symmetric matric es and henc e the original map is a submersion. Pro of that L is onto. • Both the original lo cus map and the image of L lie in the subspace of n × n symmetric matrices. • Indeed, giv en U and an y symmetric matrix S we can ﬁnd δ U such that δ U T K U + U T K δ U = S . Namely δ U = K − 1 U − T S/ 2 . • Th us, the linearization map L is onto the space of n × n of symmetric matrices and the original lo cus map U → U K U T − K to the space of symmetric matrices is a submersion. F or a submersion, the dimension of the lev el set is the dimension of the domain minus the dimension of the range space. In this case, this dimension is n 2 − n ( n + 1) / 2 = n ( n − 1) / 2. D. D. Holm Imp erial College London Applications of P oisson Geometry 28 Corollary 7.9 ( S is a matrix Lie group) S is b oth a sub gr oup and a submanifold of the gener al line ar gr oup GL ( n, R ) . Thus, by deﬁnition 7.3, S is a matrix Lie gr oup. Exercise 7.10 What is the tangent sp ac e to S at the identity, T I S ? Exercise 7.11 Show that for any p air of matric es A, B ∈ T I S , the matrix c ommutator [ A, B ] ≡ AB − B A ∈ T I S . Prop osition 7.12 The line ar sp ac e of matric es A satisfying A T K + K A = 0 deﬁnes T I S , the tangent sp ac e at the identity of the matrix Lie gr oup S deﬁne d in Pr op osition 7.5. Pro of. Near the identit y the deﬁning condition for S expands to ( I + A T + O (  2 )) K ( I + A + O (  2 )) = K , for   1 . A t linear order O (  ) one ﬁnds, A T K + K A = 0 . This relation deﬁnes the linear space of matrices A ∈ T I S . If A, B ∈ T I S , do es it follo w that [ A, B ] ∈ T I S ? Using [ A, B ] T = [ B T , A T ], w e chec k closure b y a direct computation, [ B T , A T ] K + K [ A, B ] = B T A T K − A T B T K + K AB − K B A = B T A T K − A T B T K − A T K B + B T K A = 0 . Hence, the tangen t space of S at the identit y T I S is closed under the matrix comm utator [ · , · ]. Remark 7.13 In a moment, we wil l show that the matrix c ommutator for T I S also satisﬁes the Jac obi identity. This wil l imply that the c ondition A T K + K A = 0 deﬁnes a matrix Lie algebr a. 7.2 Deﬁning Matrix Lie Algebras W e are ready to prov e the following, in preparation for deﬁning matrix Lie algebras. Prop osition 7.14 L et S b e a matrix Lie gr oup, and let A, B ∈ T I S (the tangent sp ac e to S at the identity element). Then AB − B A ∈ T I S . The pro of mak es use of a lemma. Lemma 7.15 L et R b e an arbitr ary element of a matrix Lie gr oup S , and let B ∈ T I S . Then RB R − 1 ∈ T I S . Pro of of lemma. Let R B ( t ) b e a curv e in S suc h that R B (0) = I and R 0 (0) = B . Deﬁne S ( t ) = RR B ( t ) R − 1 ∈ T I S for all t . Then S (0) = I and S 0 (0) = RB R − 1 . Hence, S 0 (0) ∈ T I S , thereby pro ving the lemma. D. D. Holm Imp erial College London Applications of P oisson Geometry 29 Pro of of Prop osition 7.14. Let R A ( s ) b e a curv e in S suc h that R A (0) = I and R 0 A (0) = A . Deﬁne S ( t ) = R A ( t ) B R A ( t ) − 1 ∈ T I S . Then the lemma implies that S ( t ) ∈ T I S for every t . Hence, S 0 ( t ) ∈ T I S , and in particular, S 0 (0) = AB − B A ∈ T I S . Deﬁnition 7.16 (Matrix comm utator) F or any p air of n × n matric es A, B , the matrix c om- mutator is deﬁne d as [ A, B ] = AB − B A . Prop osition 7.17 (Prop erties of the matrix commutator) The matrix c ommutator has the fol- lowing two pr op erties: (i) A ny two n × n matric es A and B satisfy [ B , A ] = − [ A, B ] (This is the pr op erty of skew-symmetry.) (ii) A ny thr e e n × n matric es A , B and C satisfy [[ A, B ] , C ] + [[ B , C ] , A ] + [[ C , A ] , B ] = 0 (This is known as the Jac obi identity .) Deﬁnition 7.18 (Matrix Lie algebra) A matrix Lie algebr a g is a set of n × n matric es which is a ve ctor sp ac e with r esp e ct to the usual op er ations of matrix addition and multiplic ation by r e al numb ers (sc alars) and which is close d under the matrix c ommutator [ · , · ] . Prop osition 7.19 F or any matrix Lie gr oup S , the tangent sp ac e at the identity T I S is a matrix Lie algebr a. Pro of. This follows by proposition 7.14 and b ecause T I S is a v ector space. 7.3 Examples of matrix Lie groups Example 7.20 (The Orthogonal Group O ( n ) ) The mapping c ondition U T K U = K in Pr op osi- tion 7.5 sp e cializes for K = I to U T U = I , which deﬁnes the ortho gonal gr oup. Thus, in this c ase, S sp e cializes to O ( n ) , the gr oup of n × n ortho gonal matric es. The ortho gonal gr oup is of sp e cial inter est in me chanics. Corollary 7.21 ( O ( n ) is a matrix Lie group) By Pr op osition 7.5 the ortho gonal gr oup O ( n ) is b oth a sub gr oup and a submanifold of the gener al line ar gr oup GL ( n, R ) . Thus, by deﬁnition 7.3, the ortho gonal gr oup O ( n ) is a matrix Lie gr oup. Example 7.22 (The Sp ecial Linear Group S L ( n, R ) ) The sub gr oup of GL ( n, R ) with det( U ) = 1 is c al le d S L ( n, R ) . Example 7.23 (The Sp ecial Orthogonal Group S O ( n ) ) The sp e cial c ase of S with det( U ) = 1 and K = I is c al le d S O ( n ) . In this c ase, the mapping c ondition U T K U = K sp e cializes to U T U = I with the extr a c ondition det( U ) = 1 . D. D. Holm Imp erial College London Applications of P oisson Geometry 30 Example 7.24 (The tangen t space of S O ( n ) at the iden tity T I S O ( n ) ) The sp e cial c ase with K = I of T I S O ( n ) yields, A T + A = 0 . These ar e antisymmetric matric es. Lying in the tangent sp ac e at the identity of a matrix Lie gr oup, this line ar ve ctor sp ac e forms a matrix Lie algebr a. Example 7.25 (The Symplectic Group) Supp ose n = 2 l (that is, let n b e even) and c onsider the nonsingular skew-symmetric matrix J =  0 I − I 0  wher e I is the l × l identity matrix. One may verify that S p ( l ) = { U ∈ GL (2 l, R ) | U T J U = J } is a gr oup. This is c al le d the symple ctic gr oup. R e asoning as b efor e, the matrix algebr a T I S p ( l ) is deﬁne d as the set of n × n matric es A satisfying J A T + AJ = 0 . This algebr a is denote d as sp ( l ) . Example 7.26 (The Sp ecial Euclidean Group) Consider the Lie gr oup of 4 × 4 matric es of the form E ( R , v ) =  R v 0 1  wher e R ∈ S O (3) and v ∈ R 3 . This is the sp e cial Euclide an gr oup, denote d S E (3) . The sp e cial Euclide an gr oup is of c entr al inter est in me chanics sinc e it describ es the set of rigid motions and c o or dinate tr ansformations of thr e e-dimensional sp ac e. Exercise 7.27 A p oint P in R 3 under go es a rigid motion asso ciate d with E ( R 1 , v 1 ) fol lowe d by a rigid motion asso ciate d with E ( R 2 , v 2 ) . What matrix element of S E (3) is asso ciate d with the c omp osition of these motions in the given or der? Exercise 7.28 Multiply the sp e cial Euclide an matric es of S E (3) . Investigate their matrix c ommuta- tors in their tangent sp ac e at the identity. (This is an example of a semidir e ct pr o duct Lie gr oup.) Exercise 7.29 (T rip os question) When do es a stone at the e quator of the Earth weigh the most? Two hints: (1) Assume the Earth’s orbit is a cir cle ar ound the Sun and ignor e the de clination of the Earth’s axis of r otation. (2) This is an exer cise in using S E (2) . Exercise 7.30 Supp ose the n × n matric es A and M satisfy AM + M A T = 0 . Show that exp( At ) M exp( A T t ) = M for al l t . Hint: A n M = M ( − A T ) n . This dir e ct c alculation shows that for A ∈ so ( n ) or A ∈ sp ( l ) , we have exp( At ) ∈ S O ( n ) or exp( At ) ∈ S p ( l ) , r esp e ctively. 7.4 Lie group actions The action of a Lie group G on a manifold M is a group of transformations of M asso ciated to elemen ts of the group G , whose comp osition acting on M is corresp onds to group m ultiplication in G . Deﬁnition 7.31 L et M b e a manifold and let G b e a Lie gr oup. A left action of a Lie gr oup G on M is a smo oth mapping Φ : G × M → M such that D. D. Holm Imp erial College London Applications of P oisson Geometry 31 (i) Φ( e, x ) = x for al l x ∈ M , (ii) Φ( g , Φ( h, x )) = Φ( g h, x ) for al l g , h ∈ G and x ∈ M , and (iii) Φ( g , · ) is a diﬀe omorphism on M for e ach g ∈ G . We often use the c onvenient notation g x for Φ( g , x ) and think of the gr oup element g acting on the p oint x ∈ M . The asso ciativity c ondition (ii) ab ove then simply r e ads ( gh ) x = g ( hx ) . Similarly , one can deﬁne a right action , whic h is a map Ψ : M × G → M satisfying Ψ( x, e ) = x and Ψ(Ψ( x, g ) , h ) = Ψ( x, g h ). The con venien t notation for right action is xg for Ψ( x, g ), the right action of a group element g on the p oint x ∈ M . Asso ciativit y Ψ(Ψ( x, g ) , h ) = Ψ( x, g h ) is then b e expressed con venien tly as ( xg ) h = x ( g h ). Example 7.32 (Prop erties of group actions) The action Φ : G × M → M of a gr oup G on a manifold M is said to b e: 1. tr ansitive , if for every x, y ∈ M ther e exists a g ∈ G , such that g x = y ; 2. fr e e , if it has no ﬁxe d p oints, that is, Φ g ( x ) = x implies g = e ; and 3. pr op er , if whenever a c onver gent subse quenc e { x n } in M exists, and the mapping g n x n c onver ges in M , then { g n } has a c onver gent subse quenc e in G . Orbits. Given a group action of G on M , for a giv en p oin t x ∈ M , the subset Orb x = { g x | g ∈ G } ⊂ M , is called the gr oup orbit through x . In ﬁnite dimensions, it can b e sho wn that group orbits are alw ays smo oth (p ossibly immersed) manifolds. Group orbits generalize the notion of orbits of a dynamical system. Exercise 7.33 The ﬂow of a ve ctor ﬁeld on M c an b e thought of as an action of R on M . Show that in this c ase the gener al notion of gr oup orbit r e duc es to the familiar notion of orbit use d in dynamic al systems. Theorem 7.34 Orbits of pr op er gr oup actions ar e emb e dde d submanifolds. This theorem is stated in Chapter 9 of [MaRa1994], who refer to [AbMa1978] for the pro of. Example 7.35 (Orbits of S O (3) ) A simple example of a gr oup orbit is the action of S O (3) on R 3 given by matrix multiplic ation: The action of A ∈ S O (3) on a p oint x ∈ R 3 is simply the pr o duct A x . In this c ase, the orbit of the origin is a single p oint (the origin itself ), while the orbit of any other p oint is the spher e thr ough that p oint. Example 7.36 (Orbits of a Lie group acting on itself ) The action of a gr oup G on itself fr om either the left, or the right, also pr o duc es gr oup orbits. This action sets the stage for discussing the tangent lifte d action of a Lie gr oup on its tangent bund le. L eft and right tr anslations on the gr oup ar e denote d L g and R g , r esp e ctively. F or example, L g : G → G is the map given by h → g h , while R g : G → G is the map given by h → hg , for g , h ∈ G . D. D. Holm Imp erial College London Applications of P oisson Geometry 32 (a) L eft tr anslation L g : G → G ; h → g h deﬁnes a tr ansitive and fr e e action of G on itself. R ight multiplic ation R g : G → G ; h → hg deﬁnes a right action, while h → hg − 1 deﬁnes a left action of G on itself. (b) G acts on G by c onjugation, g → I g = R g − 1 ◦ L g . The map I g : G → G given by h → g hg − 1 is the inner automorphism asso ciate d with g . Orbits of this action ar e c al le d c onjugacy classes . (c) Diﬀer entiating c onjugation at e gives the adjoint action of G on g : Ad g := T e I g : T e G = g → T e G = g . Explicitly, the adjoint action of G on g is given by Ad : G × g → g , Ad g ( ξ ) = T e ( R g − 1 ◦ L g ) ξ We have alr e ady se en an example of adjoint action for matrix Lie gr oups acting on matrix Lie algebr as, when we deﬁne d S ( t ) = R A ( t ) B R A ( t ) − 1 ∈ T I S as a key step in the pr o of of Pr op osition 7.14. (d) The c o adjoint action of G on g ∗ , the dual of the Lie algebr a g of G , is deﬁne d as fol lows. L et Ad ∗ g : g ∗ → g ∗ b e the dual of Ad g , deﬁne d by h Ad ∗ g α, ξ i = h α, Ad g ξ i for α ∈ g ∗ , ξ ∈ g and p airing h· , ·i : g ∗ × g → R . Then the map Φ ∗ : G × g ∗ → g ∗ given by ( g , α ) 7→ Ad ∗ g − 1 α is the c o adjoint action of G on g ∗ . 7.5 Examples: S O (3) , S E (3) , etc. A basis for the matrix Lie algebra so (3) and a map to R 3 The Lie algebra of S O ( n ) is called so ( n ). A basis ( e 1 , e 2 , e 3 ) for so (3) when n = 3 is given b y ˆ x = " 0 − z y z 0 − x − y x 0 # = xe 1 + y e 2 + z e 3 Exercise 7.37 Show that [ e 1 , e 2 ] = e 3 and cyclic p ermutations, while al l other matrix c ommutators among the b asis elements vanish. Example 7.38 (The isomorphism betw een so (3) and R 3 ) The pr evious e quation may b e written e quivalently by deﬁning the hat-op er ation ˆ ( · ) as ˆ x ij =  ij k x k , wher e ( x 1 , x 2 , x 3 ) = ( x, y , z ) . Her e  123 = 1 and  213 = − 1 , with cyclic p ermutations. The total ly antisymmetric tensor  ij k = −  j ik = −  ikj also deﬁnes the cr oss pr o duct of ve ctors in R 3 . Conse quently, we may write, ( x × y ) i =  ij k x j y k = ˆ x ij y j , that is, x × y = ˆ xy Exercise 7.39 What is the analo g of the hat map so (3) 7→ R 3 for the thr e e dimensional Lie algebr as sp (2 , R ) , so (2 , 1) , su (1 , 1) , or sl (2 , R ) ? Bac kground reading for this lecture is Chapter 9 of [MaRa1994]. D. D. Holm Imp erial College London Applications of P oisson Geometry 33 Compute the Adjoin t and adjoin t op erations b y diﬀeren tiation 1. Diﬀeren tiate I g ( h ) wrt h at h = e to pro duce the A djoint op er ation Ad : G × g → g : Ad g η = T e I g η 2. Diﬀeren tiate Ad g η wrt g at g = e in the direction ξ to get the Lie br acket [ ξ , η ] : g × g → g and thereb y to pro duce the adjoint op er ation T e (Ad g η ) ξ = [ ξ , η ] = ad ξ η Compute the co-Adjoin t and coadjoin t op erations b y taking duals 1. Ad ∗ g : g ∗ → g ∗ , the dual of Ad g , is deﬁned b y h Ad ∗ g α, ξ i = h α, Ad g ξ i for α ∈ g ∗ , ξ ∈ g and pairing h· , ·i : g ∗ × g → R . The map Φ ∗ : G × g ∗ → g ∗ giv en by ( g , α ) 7→ Ad ∗ g − 1 α deﬁnes the c o-A djoint action of G on g ∗ . 2. The pairing h ad ∗ ξ α, η i = h α, ad ξ η i deﬁnes the c o adjoint action of g on g ∗ , for α ∈ g ∗ and ξ , η ∈ g . See Chapter 9 of [MaRa1994] for more discussion of the Ad and ad op erations. Example: the rotation group S O (3) The Lie algebra so (3) and its dual. The sp ecial orthogonal group is deﬁned b y S O (3) := { A | A a 3 × 3 orthogonal matrix , det( A ) = 1 } . Its Lie algebra so (3) is formed b y 3 × 3 sk ew symmetric matrices, and its dual is denoted so (3) ∗ . The Lie algebra isomorphism ˆ : ( so (3) , [ · , · ]) → ( R 3 , × ) The Lie algebra ( so (3) , [ · , · ]), where [ · , · ] is the comm utator brack et of matrices, is isomorphic to the Lie algebra ( R 3 , × ), where × denotes the vector product in R 3 , b y the isomorphism u := ( u 1 , u 2 , u 3 ) ∈ R 3 7→ ˆ u :=   0 − u 3 u 2 u 3 0 − u 1 − u 2 u 1 0   ∈ so (3) , that is, ˆ u ij := −  ij k u k Equiv alently , this isomorphism is given b y ˆ uv = u × v for all u , v ∈ R 3 . The follo wing formulas for u , v , w ∈ R 3 ma y b e easily v eriﬁed: ( u × v ) ˆ = [ ˆ u , ˆ v ] [ ˆ u , ˆ v ] w = ( u × v ) × w u · v = − 1 2 trace( ˆ u ˆ v ) . D. D. Holm Imp erial College London Applications of P oisson Geometry 34 Ad action of S O (3) on so (3) The corresp onding adjoin t action of S O (3) on so (3) may be obtained as follo ws. F or S O (3) w e ha ve I A ( B ) = AB A − 1 . Diﬀerentiating B ( t ) at B (0) = I d gives Ad A ˆ v = d dt    t =0 AB ( t ) A − 1 = A ˆ v A − 1 , with ˆ v = B 0 (0) . One calculates the pairing with a v ector w ∈ R 3 as Ad A ˆ v ( w ) = A ˆ v ( A − 1 w ) = A ( v × A − 1 w ) = A v × w = ( A v )ˆ w where w e hav e used a relation A ( u × v ) = A u × A v whic h holds for any u , v ∈ R 3 and A ∈ S O (3). Consequen tly , Ad A ˆ v = ( A v ) ˆ Iden tifying so (3) ' R 3 then giv es Ad A v = A v . So (sp eaking prose all our liv es) the adjoint action of S O (3) on so (3) may be iden titiﬁed with multi- plication of a matrix in S O (3) times a vector in R 3 . ad-action of so (3) on so (3) Diﬀeren tiating again giv es the ad-action of the Lie algebra so (3) on itself: [ ˆ u , ˆ v ] = ad ˆ u ˆ v = d dt     t =0  e t ˆ u v  ˆ = ( ˆ uv ) ˆ = ( u × v ) ˆ . So in this isomorphism the v ector cross pro duct is iden tiﬁed with the matrix commutator of skew symmetric matrices. Inﬁnitesimal generator Lik ewise, the inﬁnitesimal gener ator corresponding to u ∈ R 3 has the expression u R 3 ( x ) := d dt     t =0 e t ˆ u x = ˆ u x = u × x . Exercise 7.40 What is the analo g of the hat map so (3) 7→ R 3 for the thr e e dimensional Lie algebr as sp (2 , R ) , so (2 , 1) , su (1 , 1) , or sl (2 , R ) ? The dual Lie algebra isomorphism ˜ : so (3) ∗ → R 3 Coadjoin t actions The dual so (3) ∗ is iden tiﬁed with R 3 b y the isomorphism Π ∈ R 3 7→ ˜ Π ∈ so (3) ∗ : ˜ Π ( ˆ u ) := Π · u for any u ∈ R 3 . In terms of this isomorphism, the co-Adjoin t action of S O (3) on so (3) ∗ is giv en by Ad ∗ A − 1 ˜ Π = ( A Π ) ˜ and the coadjoin t action of so (3) on so (3) ∗ is giv en by ad ∗ ˆ u ˜ Π = ( Π × u ) ˜ . (7.1) D. D. Holm Imp erial College London Applications of P oisson Geometry 35 Computing the co-Adjoint action of S O (3) on so (3) ∗ This is giv en by  Ad ∗ A − 1 ˜ Π  ( ˆ u ) = ˜ Π · Ad A − 1 ˆ u = ˜ Π · ( A − 1 u ) ˆ = Π · A T u = A Π · u = ( A Π ) ˜ ( ˆ u ) , that is, the co-Adjoin t action of S O (3) on so (3) ∗ has the expression Ad ∗ A − 1 ˜ Π = ( A Π ) ˜ , Therefore, the co-Adjoint orbit O = { A Π | A ∈ S O (3) } ⊂ R 3 of S O (3) through Π ∈ R 3 is a 2-sphere of radius k Π k . Computing the coadjoint action of so (3) on so (3) ∗ Let u , v ∈ R 3 and note that D ad ∗ ˆ u ˜ Π , ˆ v E = D ˜ Π , [ ˆ u , ˆ v ] E = D ˜ Π , ( u × v ) ˆ E = Π · ( u × v ) = ( Π × u ) · v = h Π × u ) ˜ , ˆ v i , whic h shows that ad ∗ ˆ u ˜ Π = ( Π × u ) ˜ , thereby pro ving (7.1). Therefore, T Π O =  Π × u | u ∈ R 3  , since the plane p erpendicular to Π , that is, the tangen t space to the sphere centered at the origin of radius k Π k , is giv en by  Π × u | u ∈ R 3  . 8 Lifted Actions Deﬁnition 8.1 L et Φ : G × M → M b e a left action, and write Φ g ( x ) = Φ( g , x ) for x ∈ M . The tangent lift action of G on the tangent bund le T M is deﬁne d by g v = T x Φ g ( v ) for every v ∈ T x M . Remark 8.2 In standar d c alculus notation, the expr ession for tangent lift may b e written as T x Φ · v = d dt Φ( c ( t ))    t =0 = ∂ Φ ∂ c c 0 ( t )    t =0 =: D Φ( x ) · v , with c (0) = x , c 0 (0) = v . Deﬁnition 8.3 If X is a ve ctor ﬁeld on M and φ is a diﬀer entiable map fr om M to itself, then the push-forwar d of X by φ is the ve ctor ﬁeld φ ∗ X deﬁne d by ( φ ∗ X ) ( φ ( x )) = T x φ ( X ( x )) . That is, the fol lowing diagr am c ommutes: - T M T M T φ 6 φ ∗ X - φ 6 M M X If φ is a diﬀe omorphism then the pul l-b ack φ ∗ X is also deﬁne d: ( φ ∗ X ) ( x ) = T φ ( x ) φ − 1 ( X ( φ ( x ))) . Deﬁnition 8.4 L et Φ : G × M → M b e a left action, and write Φ g ( m ) = Φ( g , m ) . Then G has a left action on X ∈ X ( M ) (the set of ve ctor ﬁelds on M ) by the push-forwar d: g X = (Φ g ) ∗ X . D. D. Holm Imp erial College London Applications of P oisson Geometry 36 Deﬁnition 8.5 L et G act on M on the left. A ve ctor ﬁeld X on M is invariant with r esp e ct to this action (we often say “ G -invariant” if the action is understo o d) if g X = X for al l g ∈ G ; e quivalently (using al l of the ab ove deﬁnitions!) g ( X ( x )) = X ( g x ) for al l g ∈ G and al l x ∈ X. Deﬁnition 8.6 Consider the left action of G on itself by left multiplic ation, Φ g ( h ) = L g ( h ) = gh. A ve ctor ﬁeld on G that is invariant with r esp e ct to this action is c al le d left-invariant . F r om Deﬁnition 8.5, we se e that X is left-invariant if and only if g ( X ( h )) = X ( g h ) , which in less c omp act notation me ans T h L g X ( h ) = X ( g h ) . The set of al l such ve ctor ﬁelds is written X L ( G ) . Prop osition 8.7 Given a ξ ∈ T e G, deﬁne X L ξ ( g ) = g ξ (r e c al l: g ξ ≡ T e L g ξ ). Then X L ξ is the unique left-invariant ve ctor ﬁeld such that X L ξ ( e ) = ξ . Pro of T o show that X L ξ is left-in v ariant, w e need to show that g  X L ξ ( h )  = X L ξ ( g h ) for every g , h ∈ G. This follows from the deﬁnition of X L ξ and the asso ciativit y prop ert y of group actions: g  X L ξ ( h )  = g ( hξ ) = ( g h ) ξ = X L ξ ( g h ) W e rep eat the last line in less compact notation: T h L g  X L ξ ( h )  = T h L g ( hξ ) = T e L g h ξ = X L ξ ( g h ) F or uniqueness, supp ose X is left-inv ariant and X ( e ) = ξ . Then for an y g ∈ G, we hav e X ( g ) = g ( X ( e )) = g ξ = X L ξ ( g ) .  Remark 8.8 Note that the map ξ 7→ X L ξ is an ve ctor sp ac e isomorphism fr om T e G to X L ( G ) . All of the ab o ve deﬁnitions hav e analogues for righ t actions. The deﬁnitions of right-invariant , X R ( G ) and X R ξ use the righ t action of G on itself deﬁned by Φ( g , h ) = R g ( h ) = hg . Exercise 8.9 Ther e is a left action of G on itself deﬁne d by Φ g ( h ) = hg − 1 . W e will use the map ξ 7→ X L ξ to relate the Lie brack et on g , deﬁned as [ ξ , η ] = ad ξ η , with the Jacobi-Lie brac ket on v ector ﬁelds. Deﬁnition 8.10 The Jac obi-Lie br acket on X ( M ) is deﬁne d in lo c al c o or dinates by [ X , Y ] J − L ≡ ( D X ) · Y − ( D Y ) · X which, in ﬁnite dimensions, is e quivalent to [ X , Y ] J − L ≡ − ( X · ∇ ) Y + ( Y · ∇ ) X ≡ − [ X , Y ] Theorem 8.11 [Pr op erties of the Jac obi-Lie br acket] 1. The Jac obi-Lie br acket satisﬁes [ X , Y ] J − L = L X Y ≡ d dt     t =0 Φ ∗ t Y , wher e Φ is the ﬂow of X . (This is c o or dinate-fr e e, and c an b e use d as an alternative deﬁnition.) D. D. Holm Imp erial College London Applications of P oisson Geometry 37 2. This br acket makes X L ( M ) a Lie algebr a with [ X , Y ] J − L = − [ X , Y ] , wher e [ X , Y ] is the L ie algebr a br acket on X ( M ) . 3. φ ∗ [ X , Y ] = [ φ ∗ X , φ ∗ Y ] for any diﬀer entiable φ : M → M . Remark 8.12 The ﬁrst pr op erty of the Jac obi-Lie br acket is pr ove d for matric es in se ction 9. The other two pr op erties ar e pr ove d b elow for the c ase that M is the Lie gr oup G . Theorem 8.13 X L ( G ) is a sub algebr a of X ( G ) . Pro of. Let X , Y ∈ X L ( G ) . Using the last item of the previous theorem, and then the G in v ariance of X and Y , gives the push-forward relations ( L g ) ∗ [ X , Y ] J − L = [( L g ) ∗ X , ( L g ) ∗ Y ] J − L for all g ∈ G. Hence [ X , Y ] J − L ∈ X L ( G ) . This is the second prop ert y in Theorem 8.11. Theorem 8.14 Set [ X L ξ , X L η ] J − L ( e ) = [ ξ , η ] for every ξ , η ∈ g , wher e the br acket on the right is the Jac obi-Lie br acket. (We say: the Lie br acket on g is the pul l-b ack of the Jac obi-Lie br acket by the map ξ 7→ X L ξ . ) Pro of. The pro of of this theorem for matrix Lie algebras is relatively easy: w e hav e already seen that ad A B = AB − B A. On the other hand, since X L A ( C ) = C A for all C , and this is linear in C, we ha ve D X L B ( I ) · A = AB , so [ A, B ] = [ X L A , X L B ] J − L ( I ) = D X L B ( I ) · X L A ( I ) − DX L A ( I ) · X L B ( I ) = D X L B ( I ) · A − D X L A ( I ) · B = AB − B A This is the third property of the Jacobi-Lie brac k et listed in Theorem 8.11. F or the general pro of, see Marsden and Ratiu [MaRa1994], Prop osition 9.14. Remark 8.15 This the or em, to gether with Item 2 in The or em 8.11, pr oves that the Jac obi-Lie br acket makes g into a Lie algebr a. Remark 8.16 By The or em 8.13, the ve ctor ﬁeld [ X L ξ , X L η ] is left-invariant. Sinc e [ X L ξ , X L η ] J − L ( e ) = [ ξ , η ] , it fol lows that [ X L ξ , X L η ] = X L [ ξ ,η ] . Deﬁnition 8.17 L et Φ : G × M → M b e a left action, and let ξ ∈ g . L et g ( t ) b e a p ath in G such that g (0) = e and g 0 (0) = ξ . Then the inﬁnitesimal gener ator of the action in the ξ dir e ction is the ve ctor ﬁeld ξ M on M deﬁne d by ξ M ( x ) = d dt     t =0 Φ g ( t ) ( x ) Remark 8.18 Note: this deﬁnition do es not dep end on the choic e of g ( t ) . F or example, the choic e in Marsden and R atiu [MaR a1994] is exp( tξ ) , wher e exp denotes the exp onentiation on Lie gr oups (not deﬁne d her e). Exercise 8.19 Consider the action of S O (3) on the unit spher e S 2 ar ound the origin, and let ξ = (0 , 0 , 1) ˆ . Sketch the ve ctor ﬁeld ξ M . (Hint: the ve ctors al l p oint “Eastwar d.”) D. D. Holm Imp erial College London Applications of P oisson Geometry 38 Theorem 8.20 F or any left action of G, the Jac obi-Lie br acket of inﬁnitesimal gener ators is r elate d to the Lie br acket on g as fol lows (note the minus sign): [ ξ M , η M ] = − [ ξ , η ] M F or a pro of, see Marsden and Ratiu [MaRa1994], Proposition 9.3.6. Exercise 8.21 Expr ess the statements and formulas of this le ctur e for the c ase of S O (3) action on its Lie algebr a so (3) . (Hint: lo ok at the pr evious le ctur e.) Wher ever p ossible, tr anslate these formulas to R 3 by using the ˆ map: so (3) → R 3 . Write the Lie algebr a for so (3) using the Jac obi-Lie br acket in terms of line ar ve ctor ﬁelds on R 3 . What ar e the char acteristic curves of these line ar ve ctor ﬁelds? D. D. Holm Imp erial College London Applications of P oisson Geometry 39 9 Handout: The Lie Deriv ativ e and the Jacobi-Lie Brac k et Let X and Y b e t wo v ector ﬁelds on the same manifold M . Deﬁnition 9.1 The Lie derivative of Y with r esp e ct to X is L X Y ≡ d dt Φ ∗ t Y   t =0 , wher e Φ is the ﬂow of X . The Lie deriv ative L X Y is “the deriv ativ e of Y in the direction given b y X . ” Its deﬁnition is co ordinate-independent. By con trast, D Y · X (also written as X [ Y ]) is also “the deriv ative of Y in the X direction”, but the v alue of D Y · X dep ends on the co ordinate system, and in particular do es not usually equal L X Y in the c hosen co ordinate system. Theorem 9.2 L X Y = [ X , Y ] , wher e the br acket on the right is the Jac obi-Lie br acket . Pro of. In the following calculation, w e assume that M is ﬁnite-dimensional, and we work in local co ordinates. Thus w e ma y consider everything as matrices, whic h allo ws us to use the product rule and the iden tities  M − 1  0 = − M − 1 M 0 M − 1 and d dt ( D Φ t ( x )) = D  d dt Φ t  ( x ) . L X Y ( x ) = d dt Φ ∗ t Y ( x )     t =0 = d dt ( D Φ t ( x )) − 1 Y (Φ t ( x ))     t =0 =  d dt ( D Φ t ( x )) − 1  Y (Φ t ( x )) + ( D Φ t ( x )) − 1 d dt Y (Φ t ( x ))  t =0 =  − ( D Φ t ( x )) − 1  d dt D Φ t ( x )  ( D Φ t ( x )) − 1 Y (Φ t ( x )) + ( D Φ t ( x )) − 1 d dt Y (Φ t ( x ))  t =0 =  −  d dt D Φ t ( x )  Y ( x ) + d dt Y (Φ t ( x ))  t =0 = − D  d dt Φ t ( x )     t =0  Y ( x ) + D Y ( x )  d dt Φ t ( x )     t =0  = − D X ( x ) · Y ( x ) + D Y ( x ) · X ( x ) = [ X, Y ] J − L ( x ) Therefore L X Y = [ X, Y ] J − L .  V orticity dynamics The same form ula applies in inﬁnite dimensions, although the pro of is more elab orate. F or example, the equation for the vorticit y dynamics of an Euler ﬂuid with velocity u (with div u = 0) and vorticit y ω = curl u may b e written as, ∂ t ω = − u · ∇ ω + ω · ∇ u = − [ u, ω ] = − ad u ω = − L u ω D. D. Holm Imp erial College London Applications of P oisson Geometry 40 All of these equations express the inv ariance of the vorticit y vector ﬁeld ω under the ﬂow of its corresp onding divergenceless velocity vector ﬁeld u . This is also encapsulated in the language of ﬂuid dynamics in char acteristic form as d dt  ω · ∂ ∂ x  = 0 , along d x dt = u ( x , t ) = curl − 1 ω . Here, the curl-in verse operator is deﬁned by the Biot-Sa v art Law, u = curl − 1 ω = curl( − ∆) − 1 ω , whic h follows from the iden tit y curl curl u = − ∆ u + ∇ div u , and application of div u = 0. Th us, in co ordinates, d x dt = u ( x , t ) = ⇒ x ( t, x 0 ) = Φ t x 0 with Φ 0 = I d , that is x (0 , x 0 ) = x 0 at t = 0 , and ω j ( x ( t, x 0 ) , t ) ∂ ∂ x j ( t, x 0 ) = ω A ( x 0 ) ∂ ∂ x A 0 ◦ Φ − 1 t . Consequen tly , Φ t ∗ ω j ( x ( t, x 0 ) , t ) = ω A ( x 0 ) ∂ x j ( t, x 0 ) ∂ x A 0 =: D Φ t · ω . This is the Cauc hy (1859) solution of Euler’s e quation for vorticity , ∂ ω ∂ t = [ ω , curl − 1 ω ] . This type of equation will reapp ear several more times in the rem aining lectures. In it, the vorticit y ω evolv es by the ad-action of the righ t-inv ariant v ector ﬁeld u = curl − 1 ω . That is, ∂ ω ∂ t = − ad curl − 1 ω ω . The Cauc hy solution is the tangen t lift of this ﬂo w, namely , Φ t ∗ ω (Φ t ( x 0 )) = T x 0 Φ t ( ω ( x 0 )) . D. D. Holm Imp erial College London Applications of P oisson Geometry 41 10 Handout: Summary of Euler’s equations for incompressible ﬂo w • Euler’s e quation of inc ompr essible ﬂuid motion u t + u · ∇ u | {z } d u /dt along d x /dt = u (advectiv e time deriv ative) + ∇ p = 0 where u : R 3 × R → R 3 satisﬁes div u = 0. • Ge ometric dynamics of vorticity ω = curl u ω t = − u · ∇ ω + ω · ∇ u = − [ u , ω ] = − ad u ω = −L u ω In these equations, one denotes d dt = ∂ ∂ t + L u and, hence, ma y write Euler vorticit y dynamics equiv alently in any of the follo wing three forms d ω dt = ω · ∇ u , as w ell as  ∂ ∂ t + L u   ω · ∂ ∂ x  = 0 or d dt ( ω · ∇ ) = 0 along d x dt = u The last form is found using the c hain rule as d dt ( ω · ∇ ) = d ω dt · ∇ + ω · d dt ∇ =  d dt ω − ω · ∇ u  · ∇ = 0 . • Ertel’s the or em [Er1942] The op erators d/dt and ω · ∇ commute on solutions of Euler’s ﬂuid equations. That is,  d dt , ω · ∇  = 0 , so that d dt ( ω · ∇ A ) = ω · ∇ d dt A for all diﬀeren tiable A when ω = curl u and u is a solution of Euler’s equations for incompressible ﬂuid ﬂo w. Conse- quen tly , one ﬁnds the following inﬁnite set of c onservation laws : If d A dt = 0 , then Z Φ( ω · ∇ A ) d 3 x = const for all diﬀeren tiable Φ D. D. Holm Imp erial College London Applications of P oisson Geometry 42 • Ohktani’s formula [Ohk1993] d 2 ω dt 2 = d dt ( ω · ∇ u ) = ω · ∇ d u dt = − ω · ∇∇ p = − P ω where P ij = ∂ 2 p ∂ x i ∂ x j (“Hessian” of pressure) In addition, one has the relations p = − ∆ − 1 tr( ∇ u T · ∇ u ) S = 1 2 ( ∇ u + ∇ u T ) (strain rate tensor) so that, the follo wing system of equations results, d ω dt = S ω d 2 ω dt 2 = − P ω • Kelvin (1890’s) cir culation the or em ω · ∂ ∂ x = ω j ∂ ∂ x j d u dt + ∇ p = 0 , where div u = 0, or equiv alently u j ,j = 0 in index notation. The motion equation may b e rewritten equiv alently as a 1-form relation, du i dt dx i = − dp = ∇ i p dx i along d x dt = u d dt ( u i dx i ) − u i d dt dx i | {z } = u i du i = d | u | 2 / 2 = − dp Consequen tly , d dt ( u · d x ) = − d  p − | u | 2 2  whic h b ecomes d dt I C ( u ) u · d x = − I C ( u ) d  p − u 2 2  = 0 D. D. Holm Imp erial College London Applications of P oisson Geometry 43 up on integrating around a closed lo op C ( u ) moving with velocity u The 1-form relation ab o ve ma y b e rewritten as ( ∂ t + L u )( u · d x ) = − d  p − u 2 2  whose exterior deriv ative yields using d 2 = 0 ( ∂ t + L u )( ω · d S ) = 0 where ω · d S = curl u · d S = d ( u · d x ). F or these geometric quan tities, one sees that the c haracteristic, or advectiv e deriv ativ e is equiv- alen t to a Lie deriv ativ e. Namely , d dt     advect | {z } ﬂuids = ∂ t + L u | {z } geometry • Stokes the or em The classical theorem due to Stok es I ∂ S u · d x = Z Z S curl u · d S sho ws that Kelvin’s circulation theorem is equiv alen t to conserv ation of ﬂux of v orticity d dt Z Z S ω · d S = 0 , with ∂ S = C ( u ) through an y surface comoving with the ﬂo w. Recall the deﬁnition, ω · ∂ ∂ x d 3 x = ω · d S One ma y chec k this form ula directly , by computing  ω 1 ∂ ∂ x 1 + ω 2 ∂ ∂ x 2 + ω 3 ∂ ∂ x 3   dx 1 ∧ dx 2 ∧ dx 3  = ω 1 dx 2 ∧ dx 3 + ω 2 dx 3 ∧ dx 1 ω dx 1 ∧ dx 2 = ω · d S One ma y then use the vorticit y equation in v ector-ﬁeld form, ( ∂ t + L u ) ω · ∂ ∂ x = 0 to pro ve that the ﬂux of v orticit y through any como ving surface is conserv ed, as follows. ( ∂ t + L u )  ω · ∂ ∂ x d 3 x  =  ( ∂ t + L u ) ω · ∂ ∂ x | {z } = 0  d 3 x + ω · ∂ ∂ x ( ∂ t + L u ) d 3 x | {z } = div u d 3 x = 0 = 0 That is, as computed ab o v e using the exterior deriv ativ e ( ∂ t + L u ) ω · d S = 0 . D. D. Holm Imp erial College London Applications of P oisson Geometry 44 • Momentum c onservation F rom Euler’s ﬂuid equation du i /dt + ∇ i p = 0 with u j ,j = 0 one ﬁnds, Z ( ∂ t u i + u j ∂ j u i + ∂ i p ) d 3 x = 0 = d dt Z u i d 3 x + Z ∂ j ( u i u j + p δ j i ) d 3 x = d dt M i | {z } = 0 + I b n j ( u i u j + p δ j i ) d S | {z } = 0, if b n · u = 0 Lo cal conserv ation of ﬂuid momentum is expressed using diﬀeren tiation by parts as ∂ t u i = − ∂ j T j i where T j i := u i u j + p δ j i is the ﬂuid str ess tensor . Moreo ver, each comp onen t of the total momen tum M i = R u i d 3 x for i = 1 , 2 , 3 , is conserv ed for an incompressible Euler ﬂo w, provided the ﬂow is tangen tial to any ﬁxed boundaries, i.e., b n · u = 0. • Mass c onservation F or mass density D ( x , t ) with total mass R D ( x , t ) d 3 x , along d x /dt = u ( x, t ) one ﬁnds, d dt D d 3 x = ( ∂ t + L u )( D d 3 x ) = ( ∂ t D + div D u ) | {z } con tinuit y eqn d 3 x = 0 The solution of this equation is written in Lagrangian form as ( D d 3 x ) · g − 1 ( t ) = D ( x 0 ) d 3 x F or incompressible ﬂow, this b ecomes 1 D = det ∂ x ∂ x 0 = d 3 x d 3 x 0 = 1 Lik ewise, in the Eulerian representation one ﬁnds the equiv alent relations, D = 1 ∂ t D + div( D u ) = 0  ⇒ div u = 0 • Ener gy c onservation Euler’s ﬂuid equation for incompressible ﬂo w div u = 0 ∂ t u + u · ∇ u + ∇ p = 0 conserv es the total kinetic energy , deﬁned by K E = Z 1 2 | u | 2 d 3 x D. D. Holm Imp erial College London Applications of P oisson Geometry 45 The v ector calculus identit y u · ∇ u = − u × curl u + 1 2 ∇| u | 2 recasts Euler’s equation as ∂ t u − u × curl u + ∇  p + u 2 2  = 0 So that ∂ ∂ t | u | 2 2 + div  p + | u | 2 2  u = 0 Consequen tly , d dt Z Ω | u | 2 2 d 3 x = − I ∂ Ω  p + | u | 2 2  u · d S = 0 since u · d S = u · b n dS = 0 on any ﬁxed boundary and one ﬁnds K E = Z 1 2 | u | 2 d 3 x = const for Euler ﬂuid motion. D. D. Holm Imp erial College London Applications of P oisson Geometry 46 11 Lie group action on its tangen t bundle Deﬁnition 11.1 A Lie gr oup G acts on its tangent bund le T G by tangent lifts. 3 Given X ∈ T h G we c an c onsider the action of G on X by either left or right tr anslations, denote d as T h L g X or T h R g X , r esp e ctively. These expr essions may b e abbr eviate d as T h L g X = L ∗ g X = g X and T h R g X = R ∗ g X = X g . L eft action of a Lie gr oup G on its tangent bund le T G is il lustr ate d in the ﬁgur e b elow. - T G T G T L g 6 g X - L g 6 G G X F or matrix Lie gr oups, this action is just multiplic ation on the left or right, r esp e ctively. Left- and Right-In v ariant V ector Fields. A vector ﬁeld X on G is called left-in v ariant, if for ev ery g ∈ G one has L ∗ g X = X , that is, if ( T h L g ) X ( h ) = X ( g h ) , for every h ∈ G . The commutativ e diagram for a left-inv ariant v ector ﬁeld is illustrated in the ﬁgure b elo w. - T G T G T L g 6 X - L g 6 G G X Prop osition 11.2 The set X L ( G ) of left invariant ve ctor ﬁelds on the Lie gr oup G is a sub algebr a of X ( G ) the set of al l ve ctor ﬁelds on G . Pro of. If X , Y ∈ X L ( G ) and g ∈ G , then L ∗ g [ X , Y ] = [ L ∗ g X , L ∗ g Y ] = [ X, Y ] , Consequen tly , the Lie brack et [ X , Y ] ∈ X L ( G ). Therefore, X L ( G ) is a subalgebra of X ( G ), the set of all v ector ﬁelds on G . Prop osition 11.3 The line ar maps X L ( G ) and T e G ar e isomorphic as ve ctor sp ac es. 3 Recall deﬁnition 6.3 of tangent lifts of a diﬀeren tiable manifold. D. D. Holm Imp erial College London Applications of P oisson Geometry 47 Demonstration of prop osition. F or each ξ ∈ T e G , deﬁne a v ector ﬁeld X ξ on G b y letting X ξ ( g ) = T e L g ( ξ ) . Then X ξ ( g h ) = T e L g h ( ξ ) = T e ( L g ◦ L h )( ξ ) = T h L g ( T e L h ( ξ )) = T h L g ( X ξ ( h )) , whic h sho ws that X ξ is left in v ariant. (This proposition is stated in Chapter 9 of [MaRa1994], who refer to [AbMa1978] for the full pro of.) Deﬁnition 11.4 [ Jac obi-Lie br acket of ve ctor ﬁelds ] L et g ( t ) and h ( s ) b e curves in G with g (0) = e , h (0) = e and deﬁne ve ctor ﬁelds at the identity of G by the tangent ve ctors g 0 (0) = ξ , h 0 (0) = η . Compute the line arization of the A djoint action of G on T e G as [ ξ , η ] := d dt d ds g ( t ) h ( s ) g ( t ) − 1    s =0 ,t =0 = d dt g ( t ) η g ( t ) − 1    t =0 = ξ η − η ξ . This is the Jac obi-Lie br acket of the ve ctor ﬁelds ξ and η . Deﬁnition 11.5 The Lie br acket in T e G is deﬁne d by [ ξ , η ] := [ X ξ , X η ]( e ) , for ξ , η ∈ T e G and for [ X ξ , X η ] the Jac obi-Lie br acket of ve ctor ﬁelds. This makes T e G into a Lie algebr a. Note that [ X ξ , X η ] = X [ ξ ,η ] , for al l ξ , η ∈ T e G . Deﬁnition 11.6 The ve ctor sp ac e T e G with this Lie algebr a structur e is c al le d the Lie algebr a of G and is denote d by g . If w e let ξ L ( g ) = T e L g ξ , then the Jacobi-Lie brac k et of t wo such left-in v ariant vector ﬁelds in fact giv es the Lie algebra brack et: [ ξ L , η L ]( g ) = [ ξ , η ] L ( g ) F or the right-in v arian t case, the right hand side obtains a min us sign, [ ξ R , η R ]( g ) = − [ ξ , η ] R ( g ) . The relative minus sign arises b ecause of the diﬀerence in action ( xh − 1 ) g − 1 = x ( g h ) − 1 on the righ t v ersus ( g h ) x = g ( hx ) on the left. Inﬁnitesimal Generator. In mechanics, group actions often app ear as symmetry transformations, whic h arise through their inﬁnitesimal generators, deﬁned as follows. Deﬁnition 11.7 Supp ose Φ : G × M → M is an action. F or ξ ∈ g , Φ ξ ( t, x ) : R × M → M deﬁne d by Φ ξ ( x ) = Φ(exp tξ , x ) = Φ exp tξ ( x ) is an R − action on M . In other wor ds, Φ exp tξ → M is a ﬂow on M . The ve ctor ﬁeld on M deﬁne d by 4 ξ M ( x ) = d dt    t =0 Φ exp tξ ( x ) is c al le d the inﬁnitesimal gener ator of the action c orr esp onding to ξ . The Jacobi-Lie brac ket of inﬁnitesimal generators is related to the Lie algebra brac k et as follows: [ ξ M , η M ] = − [ ξ , η ] M . See, for example, Chapter 9 of [MaRa1994] for the pro of. 4 Recall Deﬁnition 5.22 of vector ﬁelds. D. D. Holm Imp erial College London Applications of P oisson Geometry 48 12 Lie algebras as v ector ﬁelds Deﬁnition 12.1 (The ad-op eration) F or A ∈ g we deﬁne the op er ator ad A to b e the op er ator ad : g × g → g that maps B ∈ g to [ A, B ] . We write ad A B = [ A, B ] . Deﬁnition 12.2 A r epr esentation of a Lie algebr a g on a ve ctor sp ac e V is a mapping ρ fr om g to the line ar tr ansformations of V such that for A, B ∈ g and any c onstant sc alar c , ( i ) ρ ( A + cB ) = ρ ( A ) + cρ ( B ) ( ii ) ρ ([ A, B ]) = ρ ( A ) ρ ( B ) − ρ ( B ) ρ ( A ) . If the map ρ is 1-1 the r epr esentation is said to faithful . Exercise 12.3 F or a Lie algebr a g , show that the map A → ad A is a r epr esentation of the Lie algebr a g , with g itself the ve ctor sp ac e of the r epr esentation. This is c al le d the adjoint r epr esentation . Example 12.4 (V ector ﬁeld representations of Lie algebras) The Jac obi-Lie br acket of the ve c- tor ﬁelds ξ and η in deﬁnition 9.2 may b e r epr esente d in c o or dinate charts as, η = dx ds    s =0 = v ( x ) , and ξ = dx dt    t =0 = u ( x ) . The Jac obi-Lie br acket of these two ve ctor ﬁelds yields a thir d ve ctor ﬁeld, ξ η − η ξ = dη dt    t =0 − dξ ds    s =0 = dv dx dx dt    t =0 − du dx dx ds    s =0 = dv dx · u − du dx · v = u · ∇ v − v · ∇ u . Thus, the Jac obi-Lie br acket of ve ctor ﬁelds at the tangent sp ac e of the identity T e G is close d and may b e r epr esente d in c o or dinate charts by the Lie br acket (c ommutator of ve ctor ﬁelds) [ ξ , η ] := ξ η − η ξ = u · ∇ v − v · ∇ u =: [ u, v ] . This example also pro ves the follo wing Prop osition 12.5 L et X ( R n ) b e the set of ve ctor ﬁelds deﬁne d on R n . A Lie algebr a g may b e r epr esente d on c o or dinate charts by ve ctor ﬁelds X ξ = X i ξ ∂ ∂ x i ∈ X ( R n ) for e ach element ξ ∈ g . This ve ctor ﬁeld r epr esentation satisﬁes X [ ξ ,η ] = [ X ξ , X η ] wher e [ ξ , η ] ∈ g is the Lie algebr a pr o duct and [ X ξ , X η ] is the ve ctor ﬁeld c ommutator. 13 Lagrangian and Hamiltonian F orm ulations 13.1 Newton’s equations for particle motion in Euclidean space Newton ’s e quations m i ¨ q i = F i , i = 1 , . . . , N , (no sum on i ) (13.1) describ e the ac c eler ations ¨ q i of N particles with Masses m i , i = 1 , . . . , N , Euclidean p ositions q := ( q 1 , . . . , q N ) ∈ R 3 N , D. D. Holm Imp erial College London Applications of P oisson Geometry 49 in resp onse to pr escrib e d for c es , F = ( F 1 , . . . , F N ) , acting on these particles. Supp ose the forces arise from a p otential . That is, let F i ( q ) = − ∂ V ( { q } ) ∂ q i , V : R 3 N → R , (13.2) where ∂ V /∂ q i denotes the gradien t of the potential with respect to the v ariable q i . Then Newton’s equations (13.1) b ecome m i ¨ q i = − ∂ V ∂ q i , i = 1 , . . . , N . (13.3) Remark 13.1 Newton (1620) intr o duc e d the gr avitational p otential for c elestial me chanics, now c al le d the Newtonian p otential, V ( { q } ) = N X i,j =1 − Gm i m j | q i − q j | . (13.4) 13.2 Equiv alence Theorem Theorem 13.2 (Lagrangian and Hamil tonian form ulations) Newton ’s e quations in p otential form, m i ¨ q i = − ∂ V ∂ q i , i = 1 , . . . , N , (13.5) for p article motion in Euclide an sp ac e R 3 N ar e e quivalent to the fol lowing four statements: (i) The Euler-L agr ange e quations d dt  ∂ L ∂ ˙ q i  − ∂ L ∂ q i = 0 , i = 1 , . . . , N , (13.6) hold for the L agr angian L : R 6 N = { ( q , ˙ q ) | q , ˙ q ∈ R 3 N } → R , deﬁne d by L ( q , ˙ q ) := N X i =1 m i 2 k ˙ q i k 2 − V ( q ) , (13.7) with k ˙ q i k 2 = ˙ q i · ˙ q i = ˙ q j i ˙ q k i δ j k (no sum on i ). (ii) Hamilton ’s principle of stationary action , δ S = 0 , holds for the action functional (dr op- ping i ’s) S [ q ( · )] := Z b a L ( q ( t ) , ˙ q ( t )) dt . (13.8) (iii) Hamilton ’s e quations of motion , ˙ q = ∂ H ∂ p , ˙ p = − ∂ H ∂ q , (13.9) hold for the Hamiltonian r esulting fr om the L e gendr e tr ansform , H ( q , p ) := p · ˙ q ( q , p ) − L ( q , ˙ q ( q , p )) , (13.10) D. D. Holm Imp erial College London Applications of P oisson Geometry 50 wher e ˙ q ( q , p ) solves for ˙ q fr om the deﬁnition p := ∂ L ( q , ˙ q ) /∂ ˙ q . In the c ase of Newton ’s e quations in p otential form (13.5), the L agr angian in e quation (13.7) yields p i = m i ˙ q i and the r esulting Hamiltonian is (r estoring i ’s) H = N X i =1 1 2 m i k p i k 2 | {z } Kinetic ener gy + V ( q ) | {z } Potential (iv) Hamilton ’s e quations in their Poisson br acket formulation, ˙ F = { F , H } for al l F ∈ F ( P ) , (13.11) hold with Poisson br acket deﬁne d by { F , G } := N X i =1  ∂ F ∂ q i · ∂ G ∂ p i − ∂ F ∂ p i · ∂ G ∂ q i  for al l F , G ∈ F ( P ) . (13.12) W e will prov e this theorem b y pro ving a c hain of linked equiv alence relations: (13.5) ⇔ (i) ⇔ (ii) ⇔ (iii) ⇔ (iv) as prop ositions. (The symbol ⇔ means “equiv alen t to”.) Step I. Pr o of that Newton ’s e quations (13.5) ar e e quivalent to (i). Chec k by direct v eriﬁcation. Step II. Pr o of that (i) ⇔ (ii): The Euler-L agr ange e quations (13.6) ar e e quivalent to Hamilton ’s principle of stationary action . T o simplify notation, we momentarily suppress the particle index i . W e need to prov e the solutions of (13.6) are critical p oin ts δ S = 0 of the action functional S [ q ( · )] := Z b a L ( q ( t ) , ˙ q ( t )) dt , (13.13) (where ˙ q = d q ( t ) /dt ) with resp ect to v ariations on C ∞ ([ a, b ] , R 3 N ), the space of s mooth tra jectories q : [ a, b ] → R 3 N with ﬁxed endp oin ts q a , q b . In C ∞ ([ a, b ] , R 3 N ) consider a deformation q ( t, s ), s ∈ ( − ,  ),  > 0, with ﬁxed endp oin ts q a , q b , of a curve q 0 ( t ), that is, q ( t, 0) = q 0 ( t ) for all t ∈ [ a, b ] and q ( a, s ) = q 0 ( a ) = q a , q ( b, s ) = q 0 ( b ) = q b for all s ∈ ( − ,  ). Deﬁne a variation of the curv e q 0 ( · ) in C ∞ ([ a, b ] , R 3 N ) b y δ q ( · ) := d ds     s =0 q ( · , s ) ∈ T q 0 ( · ) C ∞ ([ a, b ] , R 3 N ) , and deﬁne the ﬁrst variation of S at q 0 ( t ) to b e the follo wing deriv ative: δ S := D S [ q 0 ( · )]( δ q ( · )) := d ds     s =0 S [ q ( · , s )] . (13.14) D. D. Holm Imp erial College London Applications of P oisson Geometry 51 Note that δ q ( a ) = δ q ( b ) = 0 . With these notations, Hamilton ’s principle of stationary action states that the curve q 0 ( t ) satisﬁes the Euler-Lagrange equations (13.6) if and only if q 0 ( · ) is a critical p oin t of the action functional, that is, D S [ q 0 ( · )] = 0. Indeed, using the equalit y of mixed partials, in tegrating by parts, and taking in to accoun t that δ q ( a ) = δ q ( b ) = 0, one ﬁnds δ S := D S [ q 0 ( · )]( δ q ( · )) = d ds     s =0 S [ q ( · , s )] = d ds     s =0 Z b a L ( q ( t, s ) , ˙ q ( t, s )) dt = N X i =1 Z b a  ∂ L ∂ q i · δ q i ( t, s ) + ∂ L ∂ ˙ q i · δ ˙ q i  dt = − N X i =1 Z b a  d dt  ∂ L ∂ ˙ q i  − ∂ L ∂ q i  · δ q i dt = 0 for all smo oth δ q i ( t ) satisfying δ q i ( a ) = δ q i ( b ) = 0. This pro ves the equiv alence of (i) and (ii) , up on restoring particle index i in the last t wo lines. Step III. Pr o of that (ii) ⇔ (iii): Hamilton ’s principle of stationary action is e quivalent to Hamilton ’s c anonic al e quations . Deﬁnition 13.3 The c onjugate momenta for the L agr angian in (13.7) ar e deﬁne d as p i := ∂ L ∂ ˙ q i = m i ˙ q i ∈ R 3 , i = 1 , . . . , N , (no sum on i ) (13.15) Deﬁnition 13.4 The Hamiltonian is deﬁne d via the change of variables ( q , ˙ q ) 7→ ( q , p ) , c al le d the L e gendr e tr ansform , H ( q , p ) : = p · ˙ q ( q , p ) − L ( q , ˙ q ( q , p )) = N X i =1 m i 2 k ˙ q i k 2 + V ( q ) = N X i =1 1 2 m i k p i k 2 | {z } Kinetic ener gy + V ( q ) | {z } Potential (13.16) Remark 13.5 The value of the Hamiltonian c oincides with the total ener gy of the system. This value wil l b e shown momentarily to r emain c onstant under the evolution of Euler-L agr ange e quations (13.6). Remark 13.6 The Hamiltonian H may b e obtaine d fr om the L e gendr e tr ansformation as a function of the variables ( q , p ) , pr ovide d one may solve for ˙ q ( q , p ) , which r e quir es the L agr angian to b e r e gular , e.g., det ∂ 2 L ∂ ˙ q i ∂ ˙ q i 6 = 0 (no sum on i ) . Lagrangian (13.7) is regular and the deriv atives of the Hamiltonian may be shown to satisfy , ∂ H ∂ p i = 1 m i p i = ˙ q i = d q i dt and ∂ H ∂ q i = ∂ V ∂ q i = − ∂ L ∂ q i . D. D. Holm Imp erial College London Applications of P oisson Geometry 52 Consequen tly , the Euler-Lagrange equations (13.6) imply ˙ p i = d p i dt = d dt  ∂ L ∂ ˙ q i  = ∂ L ∂ q i = − ∂ H ∂ q i . These calculations show that the Euler-L agr ange e quations (13.6) ar e e quivalent to Hamilton ’s c anonic al e quations ˙ q i = ∂ H ∂ p i , ˙ p i = − ∂ H ∂ q i , (13.17) where ∂ H /∂ q i , ∂ H /∂ p i ∈ R 3 are the gradien ts of H with resp ect to q i , p i ∈ R 3 , resp ectiv ely . This pro ves the equiv alence of (ii) and (iii) . Remark 13.7 The Euler-L agr ange e quations ar e se c ond or der and they determine curves in c on- ﬁgur ation sp ac e q i ∈ C ∞ ([ a, b ] , R 3 N ) . In c ontr ast, Hamilton ’s e quations ar e ﬁrst or der and they determine curves in phase sp ac e ( q i , p i ) ∈ C ∞ ([ a, b ] , R 6 N ) , a sp ac e whose dimension is twic e the dimension of the c onﬁgur ation sp ac e. Step IV. Pr o of that (iii) ⇔ (iv): Hamilton ’s c anonic al e quations may b e written using a Poisson br acket. By the c hain rule and (13.17) any F ∈ F ( P ) satisﬁes, dF dt = N X i =1  ∂ F ∂ q i · ˙ q i + ∂ F ∂ p i · ˙ p i  = N X i =1  ∂ F ∂ q i · ∂ H ∂ p i − ∂ F ∂ p i · ∂ H ∂ q i  = { F , H } . This ﬁnishes the pro of of the theorem, b y pro ving the equiv alence of (iii) and (iv) . Remark 13.8 (Energy conserv ation) Sinc e the Poisson br acket is skew symmetric, { H , F } = − { F , H } , one ﬁnds that ˙ H = { H , H } = 0 . Conse quently, the value of the Hamiltonian is pr eserve d by the evolution. Thus, the Hamiltonian is said to b e a c onstant of the motion . Exercise 13.9 Show that the Poisson br acket is biline ar, skew symmetric, satisﬁes the Jac obi identity and acts as derivation on pr o ducts of functions in phase sp ac e. Exercise 13.10 Given two c onstants of motion, what do es the Jac obi identity imply ab out additional c onstants of motion? Exercise 13.11 Compute the Poisson br ackets among J i =  ij k p j q k in Euclide an sp ac e. What Lie algebr a do these Poisson br ackets r e c al l to you? Exercise 13.12 V erify that Hamiltons e quations determine d by the function h J ( z ) , ξ i = ξ · ( q × p ) give inﬁnitesimal r otations ab out the ξ − axis. D. D. Holm Imp erial College London Applications of P oisson Geometry 53 14 Hamilton’s principle on manifolds Theorem 14.1 (Hamilton’s Principle of Stationary Action) L et the smo oth function L : T Q → R b e a L agr angian on T Q . A C 2 curve c : [ a, b ] → Q joining q a = c ( a ) to q b = c ( b ) satisﬁes the Euler- L agr ange e quations if and only if δ Z b a L ( c ( t ) , ˙ c ( t )) dt = 0 . Pro of. The meaning of the v ariational deriv ative in the statemen t is the following. Consider a family of C 2 curv es c ( t, s ) for | s | < ε satisfying c 0 ( t ) = c ( t ), c ( a, s ) = q a , and c ( b, s ) = q b for all s ∈ ( − ε, ε ). Then δ Z b a L ( c ( t ) , ˙ c ( t )) dt := d ds     s =0 Z b a L ( c ( t, s ) , ˙ c ( t, s )) dt. Diﬀeren tiating under the integral sign, working in lo cal co ordinates (co vering the curv e c ( t ) by a ﬁnite n umber of co ordinate c harts), integrating b y parts, denoting v ( t ) := d ds     s =0 c ( t, s ) , and taking in to account that v ( a ) = v ( b ) = 0, yields Z b a  ∂ L ∂ q i v i + ∂ L ∂ ˙ q i ˙ v i  dt = Z b a  ∂ L ∂ q i − d dt ∂ L ∂ ˙ q i  v i dt. This v anishes for any C 1 function v ( t ) if and only if the Euler-Lagrange equations hold. Remark 14.2 The inte gr al app e aring in this the or em S ( c ( · )) := Z b a L ( c ( t ) , ˙ c ( t )) dt is c al le d the action inte gr al . It is deﬁne d on C 2 curves c : [ a, b ] → Q with ﬁxe d endp oints, c ( a ) = q a and c ( b ) = q b . Remark 14.3 V ariational derivatives of functionals vs Lie derivatives of functions . The variational derivative of a functional S [ u ] is deﬁne d as the line arization lim  → 0 S [ u + v ] − S [ u ]  = d d     =0 S [ u + v ] = D δ S δ v , v E . Comp ar e this to the expr ession for the Lie derivative of a function. If f is a r e al value d function on a manifold M and X is a ve ctor ﬁeld on M , the Lie derivative of f along X is deﬁne d as the dir e ctional derivative L X f = X ( f ) := d f · X . If M is ﬁnite-dimensional, this is L X f = X [ f ] := d f · X = ∂ f ∂ x i X i = lim  → 0 f ( x + X ) − f ( x )  . The similarity is suggestive: Namely, the Lie derivative of a function and the variational derivative of a functional ar e b oth deﬁne d as line arizations of smo oth maps in c ertain dir e ctions. D. D. Holm Imp erial College London Applications of P oisson Geometry 54 The next theorem emphasizes the role of Lagrangian one-forms and t wo-forms in the v ariational principle. The following is a direct corollary of the previous theorem. Theorem 14.4 Given a C k L agr angian L : T Q → R for k ≥ 2 , ther e exists a unique C k − 2 map E L ( L ) : ¨ Q → T ∗ Q , wher e ¨ Q :=  d 2 q dt 2    t =0 ∈ T ( T Q )    q ( t ) is a C 2 curve in Q  is a submanifold of T ( T Q ) , and a unique C k − 1 one-form Θ L ∈ Λ 1 ( T Q ) , such that for al l C 2 variations q ( t, s ) (deﬁne d on a ﬁxe d t -interval) of q ( t, 0) = q 0 ( t ) := q ( t ) , we have δ S := d ds    s =0 S [ c ( · , s )] = D S [ q ( · )] · δ q ( · ) = Z b a E L ( L ) ( q , ˙ q , ¨ q ) · δ q dt + Θ L ( q , ˙ q ) · δ q    b a | {z } cf. No ether Thm (14.1) wher e δ q ( t ) = d ds     s =0 q ( t, s ) . D. D. Holm Imp erial College London Applications of P oisson Geometry 55 15 Summary Handout for Diﬀeren tial F orms V ector ﬁelds and 1 -forms Let M b e a manifold. In what follows, all maps may b e assumed to b e C ∞ , although that’s not necessary . A v ector ﬁeld on M is a map X : M → T M suc h that X ( x ) ∈ T x M for ev ery x ∈ M . The set of all smo oth v ector ﬁelds on M is written X ( M ) . (“Smo oth” means diﬀeren tiable or C r for some r ≤ ∞ , dep ending on con text.) A (diﬀeren tial) 1 -form on M is a map θ : M → T ∗ M such that θ ( x ) ∈ T ∗ x M for every x ∈ M . More generally , if π : E → M is a bundle, then a section of the bundle is a map ϕ : M → E such that π ◦ ϕ ( x ) = x for all x ∈ M . So a v ector ﬁeld is a section of the tangen t bundle, while a 1-form is section of the cotangen t bundle. V ector ﬁelds can added and also m ultiplied b y scalar functions k : M → R , as follo ws: ( X 1 + X 2 ) ( x ) = X 1 ( x ) + X 2 ( x ) , ( k X ) ( x ) = k ( x ) X ( x ) . Diﬀeren tial forms can added and also multiplied by scalar functions k : M → R , as follows: ( α + β ) ( x ) = α ( x ) + β ( x ) , ( k θ ) ( x ) = k ( x ) θ ( x ) . W e ha ve already deﬁned the push-forward and pull-bac k of a v ector ﬁeld. The pull-bac k of a 1-form θ on N by a map ϕ : M → N is the 1-form ϕ ∗ θ on M deﬁned by ( ϕ ∗ θ ) ( x ) · v = θ ( ϕ ( x )) · T ϕ ( v ) The push-forw ard of a 1-form α on M by a diﬀeomorphism ψ : M → N is the pull-bac k of α b y ψ − 1 . A v ector ﬁeld can b e contracted with a diﬀeren tial form, using the pairing betw een tangen t and cotangen t v ectors: ( X θ ) ( x ) = θ ( x ) · X ( x ) . Note that X θ is a map from M to R . Many b o oks write i X θ in place of X θ , and the con traction op eration is often called in terior pro duct . The diﬀeren tial of f : M → R is a 1-form d f on M deﬁned by d f ( x ) · v = d dt f ( c ( t )     t =0 for an y x ∈ M , any v ∈ T x M and an y path c ( t ) in M such that c (0) = 0 and c 0 (0) = v . The left hand side, d f ( x ) · v , means the pairing b et ween cotangent and tangen t v ectors, whic h could also b e written d f ( x )( v ) or h d f ( x ) , v i . Note: X d f = £ X f = X [ f ] Remark 15.1 d f is very similar to T f , but T f is deﬁne d for al l diﬀer entiable f : M → N , wher e as d f is only deﬁne d when N = R (in this c ourse, anyway). In this c ase, T f is a map fr om T M to T R , and T f ( v ) = d f ( x ) · v ∈ T f ( x ) R for every v ∈ T x M (we have identiﬁe d T f ( x ) R with R . ) In coordinates... Let M b e n -dimensional, and let x 1 , . . . , x n b e diﬀeren tiable lo cal coordinates for M . This means that there’s an op en subset U of M and an open subset V of R n suc h that the map ϕ : U → V deﬁned by ϕ ( x ) =  x 1 ( x ) , . . . , x n ( x )  is a diﬀeomorphism. In particular, each x i is a map from M to R , so the diﬀerential dx i is deﬁned. There is also a vector ﬁeld ∂ ∂ x i for every i, whic h is deﬁned b y ∂ ∂ x i ( x ) = d dt ϕ − 1 ( ϕ ( x ) + t e i )   t =0 , where e i is the i th standard basis v ector. Exercise 15.2 V erify that ∂ ∂ x i dx j ≡ δ i j (wher e ≡ me ans the left hand side is a c onstant function with value δ i j ) D. D. Holm Imp erial College London Applications of P oisson Geometry 56 Remark 15.3 Of c ourse, given a c o or dinate system ϕ =  x 1 , . . . , x n  , it is usual to write x =  x 1 , . . . , x n  , which me ans x is identiﬁe d with  x 1 ( x ) , . . . , x n ( x )  = ϕ ( x ) . F or ev ery x ∈ M , the v ectors ∂ ∂ x i ( x ) form a basis for T x M , so ev ery v ∈ T x M can b e uniquely ex- pressed as v = v i ∂ ∂ x i ( x ) . This expression deﬁnes the tangent-lifted coordinates x 1 , . . . , x n , v 1 , . . . v n on T M (they are lo cal coordinates, deﬁned on T U ⊂ T M ). F or every x ∈ M , the co vectors dx i ( x ) form a basis for T ∗ x M , so every α ∈ T x M can b e uniquely expressed as α = α i dx i ( x ) . This expression deﬁnes the cotangent-lifted co ordinates x 1 , . . . , x n , α 1 , . . . α n on T ∗ M (they are lo cal co ordinates, deﬁned on T ∗ U ⊂ T ∗ M ). Note that the basis  ∂ ∂ x i  is dual to the basis  dx 1 , . . . , dx n  , by the previous exercise. It follows that,  α i dx i  ·  v i ∂ ∂ x i  = α i v i (w e hav e used the summation con ven tion). In mechanics, the conﬁguration space is often called Q, and the lifted co ordinates are written: q 1 , . . . q n , ˙ q 1 , . . . , ˙ q n (on T Q ) and q 1 , . . . q n , p 1 , . . . , p n (on T ∗ Q ). Wh y the distinction b et w een subscripts and sup erscripts? This is to k eep track of how quan tities v ary if coordinates are c hanged (see next exercise). One b eneﬁt is that using the summation con ven tion gives coordinate-indep endent answers. Exercise 15.4 Consider two sets of lo c al c o or dinates q i and s i on Q, r elate d by  s 1 , . . . , s n  = ψ  q 1 , . . . , q n  . V erify that the c orr esp onding tangent lifte d c o or dinates ˙ q i and ˙ s i ar e r elate d by ˙ s i = ∂ ψ i ∂ q j ˙ q j . Note that the last e quation c an b e written as ˙ s = Dψ ( q ) ˙ q , wher e ˙ s is the c olumn ve ctor ( ˙ s 1 , . . . ˙ s n ) , and similarly for ˙ q . Do the c orr esp onding c alculation on the c otangent bund le side. Se e Deﬁnition 16.11 . The next level: T T Q, T ∗ T ∗ Q, et cetera Since T Q is a manifold, w e can consider vector ﬁelds on it, which are sections of T ( T Q ) . In co ordinates, ev ery vector ﬁeld on T T Q has the form X = a i ∂ ∂ q i + b i ∂ ∂ ˙ q i , where the a i and b i are functions of q and ˙ q . Note that the same sym b ol q i has tw o interpretations: as a coordinate on T Q and as a co ordinate on Q, so ∂ ∂ q i can mean a v ector ﬁeld T Q (as ab o v e) or on Q . The tangent lift of the bundle pro jection τ : T Q → Q is a map T τ : T T Q → T Q. If X is written in co ordinates as ab o ve, then T τ ◦ X = a i ∂ ∂ q i . A v ector ﬁeld X on T T Q is second order if T τ ◦ X ( v ) = v ; in coordinates, a i = ˙ q i . The name comes from the process of reducing of second order equations to ﬁrst order ones b y introducing new v ariables ˙ q i = dq i dt . One may also consider T ∗ T Q, T T ∗ Q and T ∗ T ∗ Q. How ev er, the subscript/sup erscript distinction is problematic here. 1 -forms The 1-forms on T ∗ Q are sections of T ∗ T ∗ Q. Giv en cotangent-lifted local co ordinates  q 1 , . . . , q n , p 1 , . . . , p n  D. D. Holm Imp erial College London Applications of P oisson Geometry 57 on T ∗ Q, the general 1-form on T ∗ Q has the form a i dq i + b i dp i , where a i and b i are functions of ( q , p ) . The canonical 1 -form on T ∗ Q is θ = p i dq i , also written in the short form p dq . Pairing θ ( q , p ) with an arbitrary tangent v ector v = a i ∂ ∂ q i + b i ∂ ∂ p i ∈ T ( q ,p ) T ∗ Q giv es h θ ( q, p ) , v i =  p i dq i , a i ∂ ∂ q i + b i ∂ ∂ p i  = p i a i =  p i dq i , a i ∂ ∂ q i  = h p, T τ ∗ ( v ) i , where τ ∗ : T ∗ Q → Q is pro jection. In the last line we hav e in terpreted q i as a coordinate on Q, whic h implies that p i dq i = p, b y deﬁnition of the co ordinates p i . Note that the last line is co ordinate-free. 2 -forms Recall that a 1-form on M , ev aluated at a p oin t x ∈ M , is a linear map from T x M to R . A 2-form on M , ev aluated at a p oin t x ∈ M , is a sk ew-symmetric bilinear form on T x M ; and the bilinear form has to v ary smo othly as x changes. (Confusingly , bilinear forms can b e skew-symmetric, symmetric or neither; diﬀer ential forms are assumed to b e skew-symmetric.) The pull-bac k of a 2-form ω on N by a map ϕ : M → N is the 2-form ϕ ∗ ω on M deﬁned b y ( ϕ ∗ ω ) ( x ) ( v , w ) = θ ( ϕ ( x )) ( T ϕ ( v ) , T ϕ ( w )) The push-forw ard of a 2-form ω on M b y a diﬀeomorphism ψ : M → N is the pull-bac k of ω by ψ − 1 . A v ector ﬁeld X can b e contracted with a 2-form ω to get a 1-form X ω deﬁned by ( X ω ) ( x )( v ) = ω ( x ) ( X ( x ) , v ) for an y v ∈ T x M . A shorthand for this is ( X ω ) ( v ) = ω ( X, v ) , or just X ω = ω ( X , · ) . The tensor product of tw o 1-forms α and β is the 2-form α ⊗ β deﬁned by ( α ⊗ β ) ( v , w ) = α ( v ) β ( w ) for all v , w ∈ T ∗ x M . The w edge pro duct of t wo 1-forms α and β is the sk ew-symmetric 2-form α ∧ β deﬁned by ( α ∧ β ) ( v , w ) = α ( v ) β ( w ) − α ( w ) β ( v ) . Exterior deriv ative The diﬀerential d f of a real-v alued function is also called the exterior deriv ative of f . In this context, real-v alued functions can b e called 0-forms. The exterior deriv ative is a linear operation from 0-forms to 1-forms that satisﬁes the Leibniz iden tity , a.k.a. the pro duct rule, d ( f g ) = f dg + g d f D. D. Holm Imp erial College London Applications of P oisson Geometry 58 The exterior deriv ative of a 1-form is an alternating 2-form, deﬁned as follows: d  a i dx i  = ∂ a i ∂ x j dx j ∧ dx i . Exterior deriv ative is a linear op eration from 1-forms to 2 forms. The following identit y is easily c heck ed: d ( d f ) = 0 for all scalar functions f . n -forms See [MaRa1994], or [Le2003], or [AbMa1978]. Unless otherwise sp eciﬁed, n -forms are assumed to b e alternating. W edge pro ducts and con tractions generalise. It is a fact that all n -forms are linear com binations of w edge pro ducts of 1-forms. Thus we can deﬁne exterior deriv ative recursively b y the prop erties d ( α ∧ β ) = dα ∧ β + ( − 1) k α ∧ dβ , for all k -forms α and all forms β , and d ◦ d = 0 In lo cal coordinates, if α = α i 1 ··· i k dx i 1 ∧ · · · ∧ dx i k (sum o ver all i 1 < · · · < i k ), then dα = ∂ α i 1 ··· i k ∂ x j dx j ∧ dx i 1 ∧ · · · ∧ dx i k The Lie deriv ative of an n -form θ in the direction of the v ector ﬁeld X is deﬁned as £ X θ = d dt ϕ ∗ t θ     t =0 , where ϕ is the ﬂo w of X . Pull-bac k commutes with the operations d, , ∧ and Lie deriv ative. Cartan’s magic formula : £ X α = d ( X α ) + X dα This lo oks ev en more magic when written using the notation i X α = X α : £ X = di X + i X d An n -form α is closed if dα = 0 , and exact if α = dβ for some β . All exact forms are closed (since d ◦ d = 0), but the conv erse is false. It is true that all closed forms are lo c al ly exact; this is the P oincar´ e Lemma . Remark 15.5 F or a survey of the b asic deﬁnitions, pr op erties, and op er ations on diﬀer ential forms, as wel l as useful of tables of r elations b etwe en diﬀer ential c alculus and ve ctor c alculus, se e, e.g., Chapter 2 of [Bl2004]. D. D. Holm Imp erial College London Applications of P oisson Geometry 59 16 Euler-Lagrange equations of manifolds In Theorem 14.4, δ S := d ds    s =0 S [ c ( · , s )] = D S [ q ( · )] · δ q ( · ) = Z b a E L ( L ) ( q , ˙ q , ¨ q ) · δ q dt + Θ L ( q , ˙ q ) · δ q    b a | {z } cf. No ether Thm (16.1) where δ q ( t ) = d ds     s =0 q ( t, s ) , the map E L : ¨ Q → T ∗ Q is called the Euler-L agr ange op er ator and its expression in local co ordinates is E L ( q , ˙ q , ¨ q ) i = ∂ L ∂ q i − d dt ∂ L ∂ ˙ q i . One understands that the formal time deriv ativ e is taken in the second summand and everything is expressed as a function of ( q , ˙ q , ¨ q ). Theorem 16.1 No ether (1918) Symmetries and Conservation L aws If the action variation in (16.1) vanishes δ S = 0 b e c ause of a symmetry tr ansformation which do es not pr eserve the end p oints and the Euler-L agr ange e quations hold, then the term marke d cf. No ether Thm must also vanish. However, vanishing of this term now is interpr ete d as a c onstant of motion. Namely, the term, A ( v , w ) := h F L ( v ) , w i , or, in c o or dinates A ( q , ˙ q , δ q ) = ∂ L ∂ ˙ q i δ q i , is c onstant for solutions of the Euler-L agr ange e quations. This r esult ﬁrst app e ar e d in No ether [No1918]. In fact, the r esult in [No1918] is mor e gener al than this. In p articular, in the PDE (Partial Diﬀer ential Equation) setting one must also include the tr ansformation of the volume element in the action prin- ciple. Se e e.g. [Ol2000] for go o d discussions of the history, fr amework and applic ations of No ether’s the or em. Exercise 16.2 Show that c onservation of ener gy r esults fr om No ether’s the or em if, in Hamil- tons principle, the variations ar e chosen as δ q ( t ) = d ds     s =0 q ( t, s ) , c orr esp onding to symmetry of the L agr angian under r ep ar ametrizations of time along the given curve q ( t ) → q ( τ ( t, s )) . The canonical Lagrangian one-form and tw o-form. The one-form Θ L , whose existence and uniqueness is guaran teed by Theorem 14.4, app ears as the b oundary term of the deriv ativ e of the action in tegral, when the endp oints of the curves on the conﬁguration manifold are free. In ﬁnite dimensions, its lo cal expression is Θ L ( q , ˙ q ) := ∂ L ∂ ˙ q i d q i  = p i ( q , ˙ q ) d q i  . D. D. Holm Imp erial College London Applications of P oisson Geometry 60 The corresp onding closed tw o-form Ω L = d Θ L obtained by taking its exterior deriv ative ma y b e expressed as Ω L := − d Θ L = ∂ 2 L ∂ ˙ q i ∂ q j d q i ∧ d q j + ∂ 2 L ∂ ˙ q i ∂ ˙ q j d q i ∧ d ˙ q j  = d p i ( q , ˙ q ) ∧ d q i  . These co eﬃcien ts ma y b e written as the 2 n × 2 n skew-symmetric matrix Ω L = A ∂ 2 L ∂ ˙ q i ∂ ˙ q j − ∂ 2 L ∂ ˙ q i ∂ ˙ q j 0 ! , (16.2) where A is the skew-symmetric n × n matrix  ∂ 2 L ∂ ˙ q i ∂ q j  −  ∂ 2 L ∂ ˙ q i ∂ q j  T . Non-degeneracy of Ω L is equiv alen t to the in vertibilit y of the matrix  ∂ 2 L ∂ ˙ q i ∂ ˙ q j  . Deﬁnition 16.3 The L e gendr e tr ansformation F L : T Q → T ∗ Q is the smo oth map ne ar the identity deﬁne d by h F L ( v q ) , w q i := d ds     s =0 L ( v q + sw q ) . In the ﬁnite dimensional case, the lo cal expression of F L is F L ( q i , ˙ q i ) =  q i , ∂ L ∂ ˙ q i  = ( q i , p i ( q , ˙ q )) . If the sk ew-symmetric matrix (16.2) is inv ertible, the Lagrangian L is said to b e r e gular . In this case, b y the implicit function theorem, F L is lo cally inv ertible. If F L is a diﬀeomorphism, L is called hyp err e gular . Deﬁnition 16.4 Given a L agr angian L , the action of L is the map A : T Q → R given by A ( v ) := h F L ( v ) , v i , or, in c o or dinates A ( q , ˙ q ) = ∂ L ∂ ˙ q i ˙ q i , (16.3) and the ener gy of L is E ( v ) := A ( v ) − L ( v ) , or, in c o or dinates E ( q , ˙ q ) = ∂ L ∂ ˙ q i ˙ q i − L ( q , ˙ q ) . (16.4) 16.1 Lagrangian vector ﬁelds and conserv ation la ws Deﬁnition 16.5 A ve ctor ﬁ eld Z on T Q is c al le d a L agr angian ve ctor ﬁeld if Ω L ( v )( Z ( v ) , w ) = h d E ( v ) , w i , for al l v ∈ T q Q , w ∈ T v ( T Q ) . Prop osition 16.6 The ener gy is c onserve d along the ﬂow of a L agr angian ve ctor ﬁeld Z . Pro of. Let v ( t ) ∈ T Q b e an in tegral curv e of Z . Skew-symmetry of Ω L implies d dt E ( v ( t )) = h d E ( v ( t )) , ˙ v ( t ) i = h d E ( v ( t )) , Z ( v ( t )) i = Ω L ( v ( t )) ( Z ( v ( t )) , Z ( v ( t ))) = 0 . Th us, E ( v ( t )) is constan t in t . D. D. Holm Imp erial College London Applications of P oisson Geometry 61 16.2 Equiv alence of dynamics for hyperregular Lagrangians and Hamiltonians Recall that a Lagrangian L is said to be hyp err e gular if its Legendre transformation F L : T Q → T ∗ Q is a diﬀeomorphism. The equiv alence betw een the Lagrangian and Hamiltonian formulations for hyperregular Lagrangians and Hamiltonians is summarized b elo w, follo wing [MaRa1994]. (a) Let L b e a h yp erregular Lagrangian on T Q and H = E ◦ ( F L ) − 1 , where E is the energy of L and ( F L ) − 1 : T ∗ Q → T Q is the inv erse of the Legendre transformation. Then the Lagrangian v ector ﬁeld Z on T Q and the Hamiltonian vector ﬁeld X H on T ∗ Q are related b y the identit y ( F L ) ∗ X H = Z. F urthermore, if c ( t ) is an integral curv e of Z and d ( t ) an in tegral curve of X H with F L ( c (0)) = d (0), then F L ( c ( t )) = d ( t ) and their integral curv es coincide on the manifold Q . That is, τ Q ( c ( t )) = π Q ( d ( t )) = γ ( t ), where τ Q : T Q → Q and π Q : T ∗ Q → Q are the canonical bundle pro jections. In particular, the pull back of the in verse Legendre transformation F L − 1 induces a one-form Θ and a closed t wo-form Ω on T ∗ Q b y Θ = ( F L − 1 ) ∗ Θ L , Ω = − d Θ = ( F L − 1 ) ∗ Ω L . In co ordinates, these are the canonical presymplectic and symplectic forms, respectively , Θ = p i d q i , Ω = − d Θ = d p i ∧ d q i . (b) A Hamiltonian H : T ∗ Q → R is said to b e hyp err e gular if the smo oth map F H : T ∗ Q → T Q , deﬁned b y h F H ( α q ) , β q i := d ds     s =0 H ( α q + sβ q ) , α q , β q ∈ T ∗ q Q, is a diﬀeomorphism. Deﬁne the action of H by G := h Θ , X H i . If H is a hyperregular Hamil- tonian then the energies of L and H and the actions of L and H are related by E = H ◦ ( F H ) − 1 , A = G ◦ ( F H ) − 1 . Also, the Lagrangian L = A − E is h yp erregular and F L = F H − 1 . (c) These constructions deﬁne a bijective corresp ondence b et ween h yp erregular Lagrangians and Hamiltonians. Remark 16.7 F or thor ough discussions of many additional r esults arising fr om the Hamilton ’s prin- ciple for hyp err e gular L agr angians se e, e.g. Chapters 7 and 8 of [MaR a1994]. Exercise 16.8 (Spherical p endulum) A p article r ol ling on the interior of a spheric al surfac e un- der gr avity is c al le d a spheric al p endulum. Write down the L agr angian and the e quations of motion for a spheric al p endulum with S 2 as its c onﬁgur ation sp ac e. Show explicitly that the L agr angian is hyp err e gular. Use the L e gendr e tr ansformation to c onvert the e quations to Hamiltonian form. Find the c onservation law c orr esp onding to angular momentum ab out the axis of gr avity by “b ar e hands” metho ds. D. D. Holm Imp erial College London Applications of P oisson Geometry 62 Exercise 16.9 (Diﬀerentially rotating frames) The L agr angian for a fr e e p article of unit mass r elative to a moving fr ame is obtaine d by setting L ( ˙ q , q , t ) = 1 2 k ˙ q k 2 + ˙ q · R ( q , t ) for a function R ( q , t ) which pr escrib es the sp ac e and time dep endenc e of the moving fr ame velo c- ity. F or example, a fr ame r otating with time-dep endent fr e quency Ω( t ) ab out the vertic al axis ˆ z is obtaine d by cho osing R ( q , t ) = q × Ω( t ) ˆ z . Calculate Θ L ( q , ˙ q ) , Ω L ( q , ˙ q ) , the Euler-L agr ange op er ator E L ( L ) ( q , ˙ q , ¨ q ) , the Hamiltonian and its c orr esp onding c anonic al e quations. Exercise 16.10 Calculate the action and the ener gy for the L agr angian in Exer cise 16.9. Deﬁnition 16.11 (Cotangent lift) Given two manifolds Q and S r elate d by a diﬀe omorphism f : Q 7→ S , the c otangent lift T ∗ f : T ∗ S 7→ T ∗ Q of f is deﬁne d by h T ∗ f ( α ) , v i = h α, T f ( v ) i (16.5) wher e α ∈ T ∗ s S , v ∈ T q Q , and s = f ( q ) . As explaine d in Chapter 6 of [MaR a1994], c otangent lifts pr eserve the action of the L agr angian L , which we write as h p , ˙ q i = h α , ˙ s i , (16.6) wher e p = T ∗ f ( α ) is the c otangent lift of α under the diﬀe omorphism f and ˙ s = T f ( ˙ q ) is the tangent lift of ˙ q under the function f , which is written in Euclide an c o or dinate c omp onents as q i → s i = f i ( q ) . Pr eservation of the action in (16.6) yields the c o or dinate r elations, (T angent lift in co ordinates) ˙ s j = ∂ f j ∂ q i ˙ q i = ⇒ p i = α k ∂ f k ∂ q i (Cotangen t lift in co ordinates) Thus, in c o or dinates, the c otangent lift is the inverse tr ansp ose of the tangent lift. Remark 16.12 The c otangent lift of a function pr eserves the induc e d action one-form, h p , dq i = h α , ds i , so it is a sour c e of (pr e-)symple ctic tr ansformations. 16.3 The classic Euler-Lagrange example: Geo desic ﬂo w An imp ortant example of a Lagrangian vector ﬁeld is the geo desic spray of a Riemannian metric. A R iemannian manifold is a smo oth manifold Q endow ed with a symmetric nondegenerate cov ariant tensor g , which is p ositive deﬁnite. Th us, on each tangen t space T q Q there is a nondegenerate deﬁnite inner pro duct deﬁned b y pairing with g ( q ). If ( Q, g ) is a Riemannian manifold, there is a natural Lagrangian on it giv en b y the kinetic ener gy K of the metric g , namely , K ( v ) := 1 2 g ( q )( v q , v q ) , D. D. Holm Imp erial College London Applications of P oisson Geometry 63 for q ∈ Q and v q ∈ T q Q . In ﬁnite dimensions, in a lo cal c hart, K ( q , ˙ q ) = 1 2 g ij ( q ) ˙ q i ˙ q j . The Legendre transformation is in this case F K ( v q ) = g ( q )( v q , · ), for v q ∈ T q Q . In coordinates, this is F K ( q , ˙ q ) =  q i , ∂ K ∂ ˙ q i  = ( q i , g ij ( q ) ˙ q j ) =: ( q i , p i ) . The Euler-Lagrange equations b ecome the ge o desic e quations for the metric g , giv en (for ﬁnite dimensional Q in a lo cal c hart) b y ¨ q i + Γ i j k ˙ q j ˙ q k = 0 , i = 1 , . . . n, where the three-index quan tities Γ h j k = 1 2 g hl  ∂ g j l ∂ q k + ∂ g kl ∂ q j − ∂ g j k ∂ q l  , with g ih g hl = δ l i , are the Christoﬀel symb ols of the Levi-Civita connection on ( Q, g ). Exercise 16.13 Explicitly c ompute the ge o desic e quation as an Euler-L agr ange e quation for the ki- netic ener gy L agr angian K ( q , ˙ q ) = 1 2 g ij ( q ) ˙ q i ˙ q j . Exercise 16.14 F or kinetic ener gy L agr angian K ( q , ˙ q ) = 1 2 g ij ( q ) ˙ q i ˙ q j with i, j = 1 , 2 , . . . , N : • Compute the momentum p i c anonic al to q i for ge o desic motion. • Perform the L e gendr e tr ansformation to obtain the Hamiltonian for ge o desic motion. • Write out the ge o desic e quations in terms of q i and its c anonic al momentum p i . • Che ck dir e ctly that Hamilton ’s e quations ar e satisﬁe d. Remark 16.15 A classic pr oblem is to determine the metric tensors g ij ( q ) for which these ge o desic e quations admit enough additional c onservation laws to b e inte gr able. Exercise 16.16 Consider the L agr angian L  ( q , ˙ q ) = 1 2 k ˙ q k 2 − 1 2  (1 − k q k 2 ) 2 for a p article in R 3 . L et γ  ( t ) b e the curve in R 3 obtaine d by solving the Euler-L agr ange e quations for L  with the initial c onditions q 0 = γ  (0) , ˙ q 0 = ˙ γ  (0) . Show that lim  → 0 γ  ( t ) is a gr e at cir cle on the two-spher e S 2 , pr ovide d that q 0 has unit length and the initial c onditions satisfy q 0 · ˙ q 0 = 0 . Remark 16.17 The L agr angian ve ctor ﬁeld asso ciate d to K is c al le d the ge o desic spr ay . Sinc e the L e gendr e tr ansformation is a diﬀe omorphism (in ﬁnite dimensions or in inﬁnite dimensions if the metric is assume d to b e str ong), the ge o desic spr ay is always a se c ond or der e quation. D. D. Holm Imp erial College London Applications of P oisson Geometry 64 16.4 Co v arian t deriv ative The v ariational approac h to geo desics reco v ers the classical form ulation using cov ariant deriv atives, as follo ws. Let X ( Q ) denote the set of vector ﬁelds on the manifold Q . The c ovariant derivative ∇ : X ( Q ) × X ( Q ) → X ( Q ) ( X , Y ) 7→ ∇ X ( Y ) , of the Levi-Civita connection on ( Q, g ) is given in lo cal charts by ∇ X ( Y ) = Γ k ij X i Y j ∂ ∂ q k + X i ∂ Y k ∂ q i ∂ ∂ q k . If c ( t ) is a curv e on Q and Y ∈ X ( Q ), the co v ariant deriv ativ e of Y along c ( t ) is deﬁned by D Y D t := ∇ ˙ c Y , or lo cally ,  D Y D t  k = Γ k ij ( c ( t )) ˙ c i ( t ) Y j ( c ( t )) + d dt Y k ( c ( t )) . A v ector ﬁeld is said to b e p ar al lel tr ansp orte d along c ( t ) if D Y D t = 0 . Th us ˙ c ( t ) is parallel transp orted along c ( t ) if and only if ¨ c i + Γ i j k ˙ c j ˙ c k = 0 . In classical diﬀerential geometry a ge o desic is deﬁned to b e a curv e c ( t ) in Q whose tangent vector ˙ c ( t ) is parallel transp orted along c ( t ). As the expression ab o v e sho ws, geo desics are integral curves of the Lagrangian v ector ﬁeld deﬁned by the kinetic energy of g . Deﬁnition 16.18 A classic al me chanic al system is given by a L agr angian of the form L ( v q ) = K ( v q ) − V ( q ) , for v q ∈ T q Q . The smo oth function V : Q → R is c al le d the p otential ener gy . The total ener gy of this system is given by E = K + V and the Euler-L agr ange e quations (which ar e always se c ond or der for a hyp err e gular L agr angian) ar e ¨ q i + Γ i j k ˙ q j ˙ q k + g il ∂ V ∂ q l = 0 , i = 1 , . . . n, wher e g ij ar e the entries of the inverse matrix of ( g ij ) . Deﬁnition 16.19 If Q = R 3 and the metric is given by g ij = δ ij , these e quations ar e Newton ’s e quations of motion (13.3) of a p article in a p otential ﬁeld which launche d our discussion in L e ctur e 10. Exercise 16.20 [Gauge invarianc e] Show that the Euler-L agr ange e quations ar e unchange d under L ( q ( t ) , ˙ q ( t )) → L 0 = L + d dt γ ( q ( t ) , ˙ q ( t )) , (16.7) for any function γ : R 6 N = { ( q , ˙ q ) | q , ˙ q ∈ R 3 N } → R . D. D. Holm Imp erial College London Applications of P oisson Geometry 65 Exercise 16.21 [Gener alize d c o or dinate the or em] Show that the Euler-L agr ange e quations ar e un- change d in form under any smo oth invertible mapping f : { q 7→ s } . That is, with L ( q ( t ) , ˙ q ( t )) = ˜ L ( s ( t ) , ˙ s ( t )) , (16.8) show that d dt  ∂ L ∂ ˙ q  − ∂ L ∂ q = 0 ⇐ ⇒ d dt ∂ ˜ L ∂ ˙ s ! − ∂ ˜ L ∂ s = 0 . (16.9) Exercise 16.22 How do the Euler-L agr ange e quations tr ansform under q ( t ) = r ( t ) + s ( t ) ? Exercise 16.23 (Other example Lagrangians) Write the Euler-L agr ange e quations, then apply the L e gendr e tr ansformation to determine the Hamiltonian and Hamilton ’s c anonic al e quations for the fol lowing L agr angians. Determine which of them ar e hyp err e gular. • L ( q , ˙ q ) =  g ij ( q ) ˙ q i ˙ q j  1 / 2 (Is it p ossible to assume that L ( q , ˙ q ) = 1 ? Why?) • L ( q , ˙ q ) = −  1 − ˙ q · ˙ q  1 / 2 • L ( q , ˙ q ) = m 2 ˙ q · ˙ q + e c ˙ q · A ( q ) , for c onstants m , c and pr escrib e d function A ( q ) . How do the Euler-L agr ange e quations for this L agr angian diﬀer fr om fr e e motion in a moving fr ame with velo city e mc A ( q ) ? Example: Charged particle in a magnetic ﬁeld. Consider a particle of charge e and mass m mo ving in a magnetic ﬁeld B , where B = ∇ × A is a giv en magnetic ﬁeld on R 3 . The Lagrangian for the motion is giv en by the “minimal coupling” prescription (ja y-dot-a y) L ( q , ˙ q ) = m 2 k ˙ q k 2 + e c A ( q ) · ˙ q , in whic h the constant c is the speed of light. The deriv atives of this Lagrangian are ∂ L ∂ ˙ q = m ˙ q + e c A =: p and ∂ L ∂ q = e c ∇ A T · ˙ q Hence, the Euler-Lagrange equations for this system are m ¨ q = e c ( ∇ A T · ˙ q − ∇ A · ˙ q ) = e c ˙ q × B (Newton’s equations for the Loren tz force) The Lagrangian L is h yp erregular, b ecause p = F L ( q , ˙ q ) = m ˙ q + e c A ( q ) has the in verse ˙ q = F H ( q , p ) = 1 m  p − e c A ( q )  . The corresp onding Hamiltonian is giv en b y the inv ertible c hange of v ariables, H ( q , p ) = p · ˙ q − L ( q , ˙ q ) = 1 2 m    p − e c A    2 . (16.10) The Hamiltonian H is hyperregular since ˙ q = F H ( q , p ) = 1 m  p − e c A  has the in verse p = F L ( q , ˙ q ) = m ˙ q + e c A . The canonical equations for this Hamiltonian reco ver Newton’s equations for the Loren tz force la w. D. D. Holm Imp erial College London Applications of P oisson Geometry 66 Example: Charged particle in a magnetic ﬁeld by the Kaluza-Klein construction. Al- though the minimal-coupling Lagrangian is not expressed as the kinetic energy of a metric, Newton’s equations for the Lorentz force law may still b e obtained as geo desic equations. This is accomplished b y susp ending them in a higher dimensional space via the Kaluza-Klein c onstruction , which pro- ceeds as follo ws. Let Q K K b e the manifold R 3 × S 1 with v ariables ( q , θ ). On Q K K in tro duce the one-form A + d θ (whic h deﬁnes a connection one-form on the trivial circle bundle R 3 × S 1 → R 3 ) and introduce the Kaluza-Klein L agr angian L K K : T Q K K ' T R 3 × T S 1 7→ R as L K K ( q , θ , ˙ q , ˙ θ ) = 1 2 m k ˙ q k 2 + 1 2    D A + d θ, ( q , ˙ q , θ , ˙ θ ) E    2 = 1 2 m k ˙ q k 2 + 1 2  A · ˙ q + ˙ θ  2 . (16.11) The Lagrangian L K K is p ositiv e deﬁnite in ( ˙ q , ˙ θ ); so it may b e regarded as the kinetic energy of a metric, the Kaluza-Klein metric on T Q K K . (This construction ﬁts the idea of U (1) gauge symmetry for electromagnetic ﬁelds in R 3 . It can b e generalized to a principal bundle with compact structure group endow ed with a connection. The Kaluza-Klein Lagrangian in this generalization leads to W ong’s equations for a color-charged particle moving in a classical Y ang-Mills ﬁeld.) The Legendre transformation for L K K giv es the momenta p = m ˙ q + ( A · ˙ q + ˙ θ ) A and π = A · ˙ q + ˙ θ . (16.12) Since L K K do es not depend on θ , the Euler-Lagrange equation d dt ∂ L K K ∂ ˙ θ = ∂ L K K ∂ θ = 0 , sho ws that π = ∂ L K K /∂ ˙ θ is conserved. The char ge is now deﬁned by e := cπ . The Hamiltonian H K K asso ciated to L K K b y the Legendre transformation (23.12) is H K K ( q , θ , p , π ) = p · ˙ q + π ˙ θ − L K K ( q , ˙ q , θ , ˙ θ ) = p · 1 m ( p − π A ) + π ( π − A · ˙ q ) − 1 2 m k ˙ q k 2 − 1 2 π 2 = p · 1 m ( p − π A ) + 1 2 π 2 − π A · 1 m ( p − π A ) − 1 2 m k p − π A k 2 = 1 2 m k p − π A k 2 + 1 2 π 2 . (16.13) On the constant level set π = e/c , the Kaluza-Klein Hamiltonian H K K is a function of only the v ariables ( q , p ) and is equal to the Hamiltonian (16.10) for charged particle motion under the Lorentz force up to an additive constant. This example provides an easy but fundamental illustration of the geometry of (Lagrangian) reduction by symmetry . The canonical equations for the Kaluza-Klein Hamiltonian H K K no w repro duce Newton’s equations for the Loren tz force law. D. D. Holm Imp erial College London Applications of P oisson Geometry 67 17 The rigid b o dy in three dimensions In the absence of external torques, Euler’s equations for rigid b ody motion are: I 1 ˙ Ω 1 = ( I 2 − I 3 )Ω 2 Ω 3 , I 2 ˙ Ω 2 = ( I 3 − I 1 )Ω 3 Ω 1 , I 3 ˙ Ω 3 = ( I 1 − I 2 )Ω 1 Ω 2 , (17.1) or, equiv alently , I ˙ Ω = I Ω × Ω , where Ω = (Ω 1 , Ω 2 , Ω 3 ) is the b ody angular velocity vector and I 1 , I 2 , I 3 are the momen ts of inertia of the rigid b ody . Question 17.1 Can these e quations – as they ar e written – b e c ast into L agr angian or Hamiltonian form in any sense? (Sinc e ther e ar e an o dd numb er of e quations, they c annot b e put into c anonic al Hamiltonian form.) W e could reformulate them as: Euler–Lagrange equations on T SO(3) or Canonical Hamiltonian equations on T ∗ SO(3), b y using Euler angles and their velocities, or their conjugate momenta. How ev er, these reformulations on T SO(3) or T ∗ SO(3) would answer a diﬀerent question for a six dimensional system. W e are in terested in these structures for the equations as given abov e. Answ er 17.2 (Lagrangian form ulation) The L agr angian answer is this: These e quations may b e expr esse d in Euler–Poinc ar´ e form on the Lie algebr a R 3 using the L agr angian l ( Ω ) = 1 2 ( I 1 Ω 2 1 + I 2 Ω 2 2 + I 3 Ω 2 3 ) = 1 2 Ω T · I Ω , (17.2) which is the (r otational) kinetic ener gy of the rigid b o dy. The Hamiltonian answer to this question wil l b e discusse d later. Prop osition 17.3 The Euler rigid b o dy e quations ar e e quivalent to the rigid b o dy action principle for a r e duc e d action δ S red = δ Z b a l ( Ω ) dt = 0 , (17.3) wher e variations of Ω ar e r estricte d to b e of the form δ Ω = ˙ Σ + Ω × Σ , (17.4) in which Σ ( t ) is a curve in R 3 that vanishes at the endp oints in time. D. D. Holm Imp erial College London Applications of P oisson Geometry 68 Pro of. Since l ( Ω ) = 1 2 h I Ω , Ω i , and I is symmetric, we obtain δ Z b a l ( Ω ) dt = Z b a h I Ω , δ Ω i dt = Z b a h I Ω , ˙ Σ + Ω × Σ i dt = Z b a  − d dt I Ω , Σ  + h I Ω , Ω × Σ i  dt = Z b a  − d dt I Ω + I Ω × Ω , Σ  dt, up on in tegrating by parts and using the endp oin t conditions, Σ ( b ) = Σ ( a ) = 0. Since Σ is otherwise arbitrary , (23.3) is equiv alent to − d dt ( I Ω ) + I Ω × Ω = 0 , whic h are Euler’s equations (17.1). Let’s deriv e this v ariational principle from the standar d Hamilton’s principle. 17.1 Hamilton’s principle for rigid b o dy motion on T SO(3) An elemen t R ∈ SO(3) gives the conﬁguration of the b o dy as a map of a r efer enc e c onﬁgur ation B ⊂ R 3 to the curren t conﬁguration R ( B ); the map R tak es a reference or lab el p oin t X ∈ B to a curren t p oin t x = R ( X ) ∈ R ( B ). When the rigid b ody is in motion, the matrix R is time-dep enden t. Thus, x ( t ) = R ( t ) X with R ( t ) a curv e parameterized by time in SO(3). The v elo cit y of a p oint of the b o dy is ˙ x ( t ) = ˙ R ( t ) X = ˙ RR − 1 ( t ) x ( t ) . Since R is an orthogonal matrix, R − 1 ˙ R and ˙ RR − 1 are sk ew matrices. Consequently , we can write (recall the hat map) ˙ x = ˙ RR − 1 x = ω × x . (17.5) This formula deﬁnes the sp atial angular velo city ve ctor ω . Th us, ω is essentially given by right translation of ˙ R to the identit y . That is, the vector ω =  ˙ RR − 1  ˆ . The corresp onding b o dy angular velo city is deﬁned by Ω = R − 1 ω , (17.6) so that Ω is the angular velocity relativ e to a b ody ﬁxed frame. Notice that R − 1 ˙ R X = R − 1 ˙ RR − 1 x = R − 1 ( ω × x ) = R − 1 ω × R − 1 x = Ω × X, (17.7) D. D. Holm Imp erial College London Applications of P oisson Geometry 69 so that Ω is given b y left translation of ˙ R to the identit y . That is, the vector Ω =  R − 1 ˙ R  ˆ . The kinetic ener gy is obtained by summing up m | ˙ x | 2 / 2 (where | · | denotes the Euclidean norm) o ver the bo dy . This yields K = 1 2 Z B ρ ( X ) | ˙ R X | 2 d 3 X , (17.8) in whic h ρ is a given mass densit y in the reference conﬁguration. Since | ˙ R X | = | ω × x | = | R − 1 ( ω × x ) | = | Ω × X | , K is a quadratic function of Ω . W riting K = 1 2 Ω T · I Ω (17.9) deﬁnes the moment of inertia tensor I , whic h, pro vided the b o dy do es not degenerate to a line, is a p ositiv e-deﬁnite (3 × 3) matrix, or b etter, a quadratic form. This quadratic form can b e diagonalized b y a change of basis; thereb y deﬁning the principal axes and momen ts of inertia. In this basis, we write I = diag ( I 1 , I 2 , I 3 ) . The function K is tak en to b e the Lagrangian of the system on T SO(3) (and by means of the Legendre transformation we obtain the corresp onding Hamiltonian description on T ∗ SO(3)). Notice that K in equation (17.8) is left (not right) inv arian t on T SO(3), since Ω =  R − 1 ˙ R  ˆ . It follo ws that the corresp onding Hamiltonian will also b e left inv arian t. In the framework of Hamilton’s principle, the relation b etw een motion in R space and motion in b ody angular velocity (or Ω ) space is as follo ws. Prop osition 17.4 The curve R ( t ) ∈ SO(3) satisﬁes the Euler-L agr ange e quations for L ( R , ˙ R ) = 1 2 Z B ρ ( X ) | ˙ R X | 2 d 3 X , (17.10) if and only if Ω ( t ) deﬁne d by R − 1 ˙ Rv = Ω × v for al l v ∈ R 3 satisﬁes Euler’s e quations I ˙ Ω = I Ω × Ω . (17.11) The pro of of this relation will illustrate how to reduce v ariational principles using their symmetry groups. By Hamilton’s principle, R ( t ) satisﬁes the Euler-Lagrange equations, if and only if δ Z L ( R , ˙ R ) dt = 0 . Let l ( Ω ) = 1 2 ( I Ω ) · Ω , so that l ( Ω ) = L ( R , ˙ R ) where the matrix R and the vector Ω are related b y the hat map, Ω =  R − 1 ˙ R  ˆ . Th us, the Lagrangian L is left SO(3)-inv arian t. That is, l ( Ω ) = L ( R , ˙ R ) = L ( e , R − 1 ˙ R ) . D. D. Holm Imp erial College London Applications of P oisson Geometry 70 T o see ho w w e should use this left-inv ariance to transform Hamilton’s principle, deﬁne the sk ew matrix ˆ Ω by ˆ Ωv = Ω × v for any v ∈ R 3 . W e diﬀerentiate the relation R − 1 ˙ R = ˆ Ω with resp ect to R to get − R − 1 ( δ R ) R − 1 ˙ R + R − 1 ( δ ˙ R ) = c δ Ω . (17.12) Let the sk ew matrix ˆ Σ b e deﬁned b y ˆ Σ = R − 1 δ R , (17.13) and deﬁne the v ector Σ by ˆ Σv = Σ × v . (17.14) Note that ˙ ˆ Σ = − R − 1 ˙ RR − 1 δ R + R − 1 δ ˙ R , so R − 1 δ ˙ R = ˙ ˆ Σ + R − 1 ˙ R ˆ Σ . (17.15) Substituting (17.15) and (17.13) in to (17.12) gives − ˆ Σ ˆ Ω + ˙ ˆ Σ + ˆ Ω ˆ Σ = c δ Ω , that is, c δ Ω = ˙ ˆ Σ + [ ˆ Ω , ˆ Σ ] . (17.16) The iden tity [ ˆ Ω , ˆ Σ ] = ( Ω × Σ ) ˆ holds b y Jacobi’s identit y for the cross pro duct and so δ Ω = ˙ Σ + Ω × Σ . (17.17) These calculations pro ve the follo wing: Theorem 17.5 F or a L agr angian which is left-invariant under SO(3), Hamilton ’s variational princi- ple δ S = δ Z b a L ( R , ˙ R ) dt = 0 (17.18) on T SO(3) is e quivalent to the r e duc e d variational principle δ S red = δ Z b a l ( Ω ) dt = 0 (17.19) with Ω =  R − 1 ˙ R  ˆ on R 3 wher e the variations δ Ω ar e of the form δ Ω = ˙ Σ + Ω × Σ , with Σ ( a ) = Σ ( b ) = 0 . Recall Theorem 17.5 : F or a L agr angian which is left-invariant under SO(3), Hamilton ’s variational principle δ S = δ Z b a L ( R , ˙ R ) dt = 0 (17.20) D. D. Holm Imp erial College London Applications of P oisson Geometry 71 on T SO(3) is e quivalent to the r e duc e d variational principle δ S red = δ Z b a l ( Ω ) dt = 0 (17.21) with Ω =  R − 1 ˙ R  ˆ on R 3 wher e the variations δ Ω ar e of the form δ Ω = ˙ Σ + Ω × Σ , with Σ ( a ) = Σ ( b ) = 0 . Reconstruction of R ( t ) ∈ S O (3) . In Theorem 17.5, Euler’s equations for the rigid b o dy I ˙ Ω = I Ω × Ω , follo w from the reduced v ariational principle (17.19) for the Lagrangian l ( Ω ) = 1 2 ( I Ω ) · Ω , (17.22) whic h is expressed in terms of the left-inv ariant time-dependent angular velocity in the b ody , Ω ∈ so (3). The b ody angular velocity Ω ( t ) yields the tangen t vector ˙ R ( t ) ∈ T R ( t ) S O (3) along the in tegral curve in the rotation group R ( t ) ∈ S O (3) b y the relation, ˙ R ( t ) = R ( t ) Ω ( t ) . This relation provides the r e c onstruction formula . It’s solution as a linear diﬀeren tial equation with time-dep enden t co eﬃcien ts yields the integral curv e R ( t ) ∈ S O (3) for the orientation of the rigid b ody , once the time dep endence of Ω ( t ) is determined from the Euler equations. 17.2 Hamiltonian F orm of rigid b o dy motion. A dynamical system on a manifold M ˙ x ( t ) = F ( x ) , x ∈ M is said to b e in Hamiltonian form , if it can b e expressed as ˙ x ( t ) = { x , H } , for H : M 7→ R , in terms of a P oisson brack et op eration, {· , ·} : F ( M ) × F ( M ) 7→ F ( M ) , whic h is bilinear, skew-symmetric and satisﬁes the Jacobi iden tit y and (usually) the Leibnitz rule. As we shall explain, reduced equations arising from group-in v ariant Hamilton’s principles on Lie groups are naturally Hamiltonian. If we L e gendr e tr ansform our reduced Lagrangian for the S O (3) left inv ariant v ariational principle (17.21) for rigid b o dy dynamics, then its simple, b eautiful and w ell-known Hamiltonian form ulation emerges . Deﬁnition 17.6 The L e gendr e tr ansformation F l : so (3) → so (3) ∗ is deﬁne d by F l (Ω) = δ l δ Ω = Π . The Legendre transformation deﬁnes the b o dy angular momentum by the v ariations of the rigid- b ody’s reduced Lagrangian with resp ect to the b o dy angular v elo cit y . F or the Lagrangian in (17.22), the R 3 comp onen ts of the b ody angular momentum are Π i = I i Ω i = ∂ l ∂ Ω i , i = 1 , 2 , 3 . (17.23) D. D. Holm Imp erial College London Applications of P oisson Geometry 72 17.3 Lie-P oisson Hamiltonian form ulation of rigid b o dy dynamics. Let h (Π) := h Π , Ω i − l (Ω) , where the pairing h· , ·i : so (3) ∗ × so (3) → R is understo o d in comp onen ts as the vector dot pro duct on R 3 h Π , Ω i := Π · Ω . Hence, one ﬁnds the exp ected expression for the rigid-bo dy Hamiltonian h = 1 2 Π · I − 1 Π := Π 2 1 2 I 1 + Π 2 2 2 I 2 + Π 2 3 2 I 3 . (17.24) The Legendre transform F l for this case is a diﬀeomorphism, so w e may solve for ∂ h ∂ Π = Ω +  Π , ∂ Ω ∂ Π  −  ∂ l ∂ Ω , ∂ Ω ∂ Π  = Ω . In R 3 co ordinates, this relation expresses the b o dy angular velocity as the deriv ative of the reduced Hamiltonian with resp ect to the bo dy angular momen tum, namely (introducing grad-notation), ∇ Π h := ∂ h ∂ Π = Ω . Hence, the reduced Euler-Lagrange equations for l ma y b e expressed equiv alently in angular momen- tum v ector comp onen ts in R 3 and Hamiltonian h as: d dt ( I Ω ) = I Ω × Ω ⇐ ⇒ ˙ Π = Π × ∇ Π h := { Π , h } . This expression suggests we in tro duce the following rigid b ody P oisson brac ket on functions of the Π ’s: { f , h } ( Π ) := − Π · ( ∇ Π f × ∇ Π h ) . (17.25) F or the Hamiltonian (17.24), one chec ks that the Euler equations in terms of the rigid-b o dy angular momen ta, ˙ Π 1 = I 2 − I 3 I 2 I 3 Π 2 Π 3 , ˙ Π 2 = I 3 − I 1 I 3 I 1 Π 3 Π 1 , ˙ Π 3 = I 1 − I 2 I 1 I 2 Π 1 Π 2 , (17.26) that is, ˙ Π = Π × Ω . (17.27) are equiv alent to ˙ f = { f , h } , with f = Π . The P oisson brack et prop osed in (17.25) is an example of a Lie Poisson br acket , which w e will show separately satisﬁes the deﬁning relations to b e a P oisson brac ket. D. D. Holm Imp erial College London Applications of P oisson Geometry 73 17.4 R 3 P oisson brack et. The rigid b ody Poisson brac k et (17.25) is a sp ecial case of the P oisson brack et for functions on R 3 , { f , h } = − ∇ c · ∇ f × ∇ h (17.28) This brac ket generates the motion ˙ x = { x , h } = ∇ c × ∇ h (17.29) F or this brack et the motion takes place along the intersections of level surfaces of the functions c and h in R 3 . In particular, for the rigid b ody , the motion takes place along intersections of angular momen tum spheres c = k x k 2 / 2 and energy ellipsoids h = x · I x . (See the cov er illustration of Marsden and Ratiu [2003].) Exercise 17.7 Consider the R 3 Poisson br acket { f , h } = − ∇ c · ∇ f × ∇ h (17.30) L et c = x T · C x b e a quadr atic form on R 3 , and let C b e the asso ciate d symmetric 3 × 3 matrix. Determine the c onditions on the quadr atic function c ( x ) so that this Poisson br acket wil l satisfy the Jac obi identity. Exercise 17.8 Find the gener al c onditions on the function c ( x ) so that the R 3 br acket { f , h } = − ∇ c · ∇ f × ∇ h satisﬁes the deﬁning pr op erties of a Poisson br acket. Is this R 3 br acket also a derivation satisfying the L eibnitz r elation for a pr o duct of functions on R 3 ? If so, why? Exercise 17.9 How is the R 3 br acket r elate d to the c anonic al Poisson br acket? Hint: r estrict to level surfac es of the function c ( x ) . Exercise 17.10 (Casimirs of the R 3 brac k et) The Casimirs (or distinguishe d functions, as Lie c al le d them) of a Poisson br acket satisfy { c, h } ( x ) = 0 , ∀ h ( x ) Supp ose the function c ( x ) is chosen so that the R 3 br acket (17.28) satisﬁes the deﬁning pr op erties of a Poisson br acket. What ar e the Casimirs for the R 3 br acket (17.28)? Why? Exercise 17.11 Show that the motion e quation ˙ x = { x , h } for the R 3 br acket (17.28) is invariant under a c ertain line ar c ombination of the functions c and h . Interpr et this invarianc e ge ometric al ly. 18 Momen tum maps The Main Idea Symmetries are often asso ciated with conserved quan tities. F or example, the ﬂow of any S O (3)- in v ariant Hamiltonian v ector ﬁeld on T ∗ R 3 conserv es angular momentum, q × p . More generally , giv en a Hamiltonian H on a phase space P , and a group action of G on P that conserv es H , there is often an asso ciated “momentum map” J : P → g ∗ that is conserved by the ﬂo w of the Hamiltonian v ector ﬁeld. Note: all group actions in this section will b e left actions until otherwise sp eciﬁed. D. D. Holm Imp erial College London Applications of P oisson Geometry 74 18.1 Hamiltonian systems on P oisson manifolds Deﬁnition 18.1 A Poisson br acket on a manifold P is a skew-symmetric biline ar op er ation on F ( P ) := C ∞ ( P , R ) satisfying the Jac obi identity and the L eibniz identity, { F G, H } = F { G, H } + { F, H } G The p air ( P , {· , ·} ) is c al le d a Poisson manifold . Remark 18.2 The L eibniz identity is sometimes not include d in the deﬁnition. Note that biline arity, skew-symmetry and the Jac obi identity ar e the axioms of a Lie algebr a. In what fol lows, a Poisson br acket is a binary op er ation that makes F ( P ) into a Lie algebr a and also satisﬁes the L eibniz identity. Exercise 18.3 Show that the classic al Poisson br acket , deﬁne d in c otangent-lifte d c o or dinates  q 1 , . . . , q N , p 1 , . . . , p N  on an 2 N -dimensional c otangent bund le T ∗ Q by { F , G } = N X i =1  ∂ F ∂ q i ∂ G ∂ p i − ∂ F ∂ p i ∂ G ∂ q i  , satisﬁes the axioms of a Poisson br acket. Show also that the deﬁnition of this br acket is indep endent of the choic e of lo c al c o or dinates  q 1 , . . . , q N  . Deﬁnition 18.4 A Poisson map b etwe en two Poisson manifolds is a map ϕ : ( P 1 , {· , ·} 1 ) → ( P 2 , {· , ·} 2 ) that pr eserves the br ackets, me aning { F ◦ ϕ, G ◦ ϕ } 1 = { F , G } 2 ◦ ϕ, for al l F , G ∈ F ( P 2 ) . Deﬁnition 18.5 An action Φ of G on a Poisson manifold ( P , { , } ) is c anonic al if Φ g is a Poisson map for every g , i.e. { F ◦ Φ g , K ◦ Φ g } = { F , K } ◦ Φ g for every F , K ∈ F ( P ) . Deﬁnition 18.6 L et ( P , {· , ·} ) b e a Poisson manifold, and let H : P → R b e diﬀer entiable. The Hamiltonian ve ctor ﬁeld for H is the ve ctor ﬁeld X H deﬁne d by X H ( F ) = { F , H } , for any F ∈ F ( P ) Remark 18.7 X H is wel l-deﬁne d b e c ause of the L eibniz identity and the c orr esp ondanc e b etwe en ve ctor ﬁelds and derivations (se e [L e2003]). Remark 18.8 X H ( F ) = £ X H F = ˙ F , the Lie derivative of F along the ﬂow of X H . The e quations ˙ F = { F , H } , c al le d “Hamilton ’s e quations”, have alr e ady app e ar e d in The or em 6.2, and ar e an e quivalent deﬁnition of X H . Exercise 18.9 Show that Hamilton ’s e quations for the classic al Poisson br acket ar e the c anonic al Hamilton ’s e quations, ˙ q i = ∂ H ∂ p i , ˙ p i = − ∂ H ∂ q i . D. D. Holm Imp erial College London Applications of P oisson Geometry 75 18.2 Inﬁnitesimal inv ariance under Hamiltonian v ector ﬁelds Let G act smo othly on P , and let ξ ∈ g . Recall (from Lecture 9) that the inﬁnitesimal generator ξ P is the v ector ﬁeld on P deﬁned by ξ P ( x ) = d dt g ( t ) x     t =0 , for some path g ( t ) in G suc h that g (0) = e and g 0 (0) = ξ . Remark 18.10 F or matrix gr oups, we c an take g ( t ) = exp ( tξ ) . This works in gener al for the exp o- nential map of an arbitr ary Lie gr oup. F or matrix gr oups, ξ P ( x ) = d dt exp( tξ ) x     t =0 = ξ x (matrix multiplic ation). Exercise 18.11 If H : P → R is G -invariant, me aning that H ( g x ) = H ( x ) for al l g ∈ G and x ∈ P, then £ ξ P H = 0 for al l ξ ∈ g . This pr op erty is c al le d inﬁnitesimal invarianc e . Example 18.12 (The momen tum map for the rotation group) Consider the c otangent bund le of or dinary Euclide an sp ac e R 3 . This is the Poisson (symple ctic) manifold with c o or dinates ( q , p ) ∈ T ∗ R 3 ' R 6 , e quipp e d with the c anonic al Poisson br acket. An element g of the r otation gr oup S O (3) acts on T ∗ R 3 ac c or ding to g ( q , p ) = ( g q , g p ) Set g ( t ) = exp( tA ) , so that d dt   t =0 g ( t ) = A and the c orr esp onding Hamiltonian ve ctor ﬁeld is X A = ( ˙ q , ˙ p ) = ( A q , A p ) wher e A ∈ so (3) is a skew-symmetric matrix. The c orr esp onding Hamiltonian e quations r e ad ˙ q = A q = ∂ J A ∂ p , ˙ p = A p = − ∂ J A ∂ q . Henc e, J A ( q , p ) = − A p · q = a i  ij k p k q j = a · q × p . for a ve ctor a ∈ R 3 with c omp onents a i , i = 1 , 2 , 3 . So the momentum map for the r otation gr oup is the angular momentum J = q × p . Example 18.13 Consider angular momentum J = q × p , deﬁne d on P = T ∗ R 3 . F or every ξ ∈ R 3 , deﬁne J ξ ( q , p ) := ξ · ( q × p ) = p · ( ξ × q ) Using Exer cise 18.9 and Example 18.10, X J ξ ( q , p ) =  ∂ J ξ ∂ p , − ∂ J ξ ∂ q  = ( ξ × q , ξ × p ) = ˆ ξ P ( q , p ) , wher e the last line is the inﬁnitesimal gener ator c orr esp onding to ˆ ξ ∈ so (3) . Now supp ose H : P → R is S O (3) -invariant. F r om Exer cise 18.11, we have £ ˆ ξ H = 0 . It fol lows that £ X H J ξ = { J ξ , H } = − { H , J ξ } = − £ X J ξ H = − £ ξ P H = 0 . Sinc e this holds for al l ξ , we have shown that J is c onserve d by the Hamiltonian ﬂow. D. D. Holm Imp erial College London Applications of P oisson Geometry 76 18.3 Deﬁning Momentum Maps In order to generalise this example, we recast it using the hat map ˆ : R 3 → so (3) and the asso ciated map ˜ :  R 3  ∗ → so (3) ∗ , and the standard iden tiﬁcation  R 3  ∗ ∼ = R 3 via the Euclidean dot product. W e consider J as a function from P to so (3) ∗ giv en by J ( q , p ) = ( q × p ) ˜ . F or any ξ = ˆ v , we deﬁne J ξ ( q , p ) = D ( q × p ) ˜ , ˆ v E = ( q × p ) · v . As b efore, we ﬁnd that X J ξ = ξ P for every ξ , and J is conserv ed by the Hamiltonian ﬂow. W e take the ﬁrst prop ert y , X J ξ = ξ P , as the general deﬁnition of a momentum map. The conserv ation of J follows by the same P oisson brac ket calculation as in the example; the result is No ether’s Theorem. Deﬁnition 18.14 A momentum map for a c anonic al action of G on P is a map J : P → g ∗ such that, for every ξ ∈ g , the map J ξ : P → R deﬁne d by J ξ ( p ) = h J ( p ) , ξ i satisﬁes X J ξ = ξ P Theorem 18.15 (No ether’s Theorem) L et G act c anonic al ly on ( P , {· , ·} ) with momentum map J. If H is G -invariant, then J is c onserve d by the ﬂow of X H . Pro of F or every ξ ∈ g , £ X H J ξ = { J ξ , H } = − { H , J ξ } = − £ X J ξ H = − £ ξ P H = 0 .  Exercise 18.16 Momentum maps ar e unique up to a choic e of a c onstant element of g ∗ on every c onne cte d c omp onent of M . Exercise 18.17 Show that the S 1 action on the torus T 2 := S 1 × S 1 given by α ( θ , φ ) = ( α + θ , φ ) is c anonic al with r esp e ct to the classic al br acket (with θ , φ in plac e of q , p ), but do esn ’t have a momentum map. Exercise 18.18 Show that the Petzval invariant for F ermat’s principle in axisymmetric, tr anslation- invariant me dia is a momentum map, T ∗ R 2 7→ sp (2 , R ) ∗ taking ( q , p ) 7→ ( X , Y , Z ) . What is its symmetry? What is its Hamiltonian ve ctor ﬁeld? Theorem 18.19 (also due to No ether) L et G act on Q, and by c otangent lifts on T ∗ Q. Then J : T ∗ Q → g ∗ deﬁne d by, for every ξ ∈ g , J ξ ( α q ) = h α q , ξ Q ( q ) i , for every α q ∈ T ∗ q Q, is a momentum map (the “standar d one”) for the G action with r esp e ct to the classic al Poisson br acket. (A pro of using symplectic forms is giv en in Marsden and Ratiu [2003].) Pro of W e need to sho w that X J ξ = ξ T ∗ Q , for every ξ ∈ g . F rom the deﬁnition of Hamiltonian v ector ﬁelds, this is equiv alent to showing that ξ T ∗ Q [ F ] = { F , J ξ } for ev ery F ∈ F ( T ∗ Q ) . W e v erify D. D. Holm Imp erial College London Applications of P oisson Geometry 77 this for ﬁnite-dimensional Q b y using cotangent-lifted local co ordinates. ∂ J ξ ∂ p ( q , p ) = ξ Q ( q ) ∂ J ξ ∂ q i ( q , p ) =  p, ∂ ∂ q i ( ξ Q ( q ))  =  p, ∂ ∂ q i  ∂ ∂ t Φ (exp( tξ )) ( q )     t =0  =  p, ∂ ∂ t  ∂ ∂ q i Φ (exp( tξ )) ( q )      t =0  = ∂ ∂ t  p, T Φ (exp( tξ )) ∂ ∂ q i ( q )      t =0 = ∂ ∂ t  T ∗ Φ (exp( tξ )) p, ∂ ∂ q i ( q )      t =0 =  − ξ T ∗ Q ( q , p ) , ∂ ∂ q i ( q )  ∂ J ξ ∂ q ( q , p ) = − ξ T ∗ Q ( q , p ) So for ev ery F ∈ F ( T ∗ Q ) , ξ T ∗ Q [ F ] = ∂ ∂ t F (exp( tξ ) q , exp( tξ ) p )     t =0 = ∂ F ∂ q ξ Q ( q ) + ∂ F ∂ p ξ T ∗ Q ( q , p ) = ∂ F ∂ q ∂ J ξ ∂ p − ∂ F ∂ p ∂ J ξ ∂ q = { F , J ξ }  Example 18.20 L et G ⊂ M n ( R ) b e a matrix gr oup, with c otangent-lifte d action on ( q , p ) ∈ T ∗ R n . F or every g ⊂ M n ( R ) , q 7→ g q . The c otangent-lifte d action is ( q , p ) 7→ ( g q , g − T p ) . Thus, writing g = exp( tξ ) , the line arization of this gr oup action yields the ve ctor ﬁeld X ξ = ( ξ q , − ξ T p ) The c orr esp onding Hamiltonian e quations r e ad ξ q = ∂ J ξ ∂ p , − ξ T p = − ∂ J ξ ∂ q This yields the momentum map J ( q , p ) given by J ξ ( q , p ) = h J ( q , p ) , ξ i = p T ξ Q ( q ) = p T ξ q . In c o or dinates, p T ξ q = p i ξ i j q j , so J ( q , p ) = q i p j . Exercise 18.21 Calculate the momentum map of the c otangent lifte d action of the gr oup of tr ansla- tions of R 3 . Solution 18.22 x ∈ R 3 acts on q ∈ R 3 by addition of ve ctors, x · ( q ) = q + x . The inﬁnitesimal gener ator is lim x → 0 d d x ( q + x ) = Id . Thus, ξ q = I d and h J k , ξ i = h ( q , p ) , ξ q i = h p , I d i = p i δ i k = p k This is also Hamiltonian with J ξ = p , so that { p , J ξ } = 0 and { q , J ξ } = I d . D. D. Holm Imp erial College London Applications of P oisson Geometry 78 Example 18.23 L et G act on itself by left multiplic ation, and by c otangent lifts on T ∗ G. We ﬁrst note that the inﬁnitesimal action on G is ξ G ( g ) = d dt exp ( tξ ) g | t =0 = T R g ξ . L et J L b e the momentum map for this action. L et α g ∈ T ∗ g G. F or every ξ ∈ g , we have h J L ( α g ) , ξ i = h α g , ξ G ( g ) i = h α g , T R g ξ i =  T R ∗ g α g , ξ  so J L ( α g ) = T R ∗ g α g . A lternatively, writing α g = T ∗ L g − 1 µ for some µ ∈ g ∗ we have J L  T ∗ L g − 1 µ  = T R ∗ g T ∗ L g − 1 µ = Ad ∗ g − 1 µ. Exercise 18.24 Show that the momentum map for the right multiplic ation action R g ( h ) = hg is J R ( α g ) = T L ∗ g α g . F or matrix groups, the tangent lift of the left (or right) multiplication action is again matrix m ultiplication. Indeed, to compute T R G ( A ) for any A ∈ T Q S O (3) , let B ( t ) b e a path in S O (3) such that B (0) = Q and B 0 (0) = A. Then T R G ( A ) = d dt B ( t ) G     t =0 = AG. Similarly , T L G ( A ) = GA. T o compute the cotangent lift similarly , we need to b e able to consider elemen ts of T ∗ G as matrices. This can b e done using an y nondegenerate bilinear form on eac h tangent space T Q G. W e will use the pairing deﬁned by hh A, B ii := − 1 2 tr  A T B  = − 1 2 tr  AB T  . (The equiv alence of the tw o formulas follows from the properties tr ( C D ) = tr ( D C ) and tr ( C T ) = tr ( C )). Exercise 18.25 Che ck that this p airing, r estricte d to so (3) , c orr esp onds to the Euclide an inner pr o duct via the hat map. Example 18.26 Consider the pr evious example for a matrix gr oup G. F or any Q ∈ G, the p airing given ab ove al lows use to c onsider any element P ∈ T ∗ Q G as a matrix. The natur al p airing of T ∗ Q G with T Q G now has the formula, h P , A i = − 1 2 tr  P T A  , for al l A ∈ T Q G. We c ompute the c otangent-lifts of the left and right multiplic ation actions : h T ∗ L Q ( P ) , A i = h P , T L Q ( A ) i = h P , QA i = − 1 2 tr  P T QA  = − 1 2 tr   Q T P  T A  =  Q T P , A  h T ∗ R Q ( P ) , A i = h P , T R Q ( A ) i = h P , AQ i = − 1 2 tr  P ( AQ ) T  = − 1 2 tr  P Q T A T  =  P Q T , A  D. D. Holm Imp erial College London Applications of P oisson Geometry 79 In summary, T ∗ L Q ( P ) = Q T P and T ∗ R Q ( P ) = P Q T We thus c ompute the momentum maps as J L ( Q, P ) = T ∗ R Q P = P Q T J R ( Q, P ) = T ∗ L Q P = Q T P In the sp e cial c ase of G = S O (3) , these matric es P Q T and Q T P ar e skew-symmetric, sinc e they ar e elements of so (3) . Ther efor e, J L ( Q, P ) = T ∗ R Q P = 1 2  P Q T − QP T  J R ( Q, P ) = T ∗ L Q P = 1 2  Q T P − P T Q  Exercise 18.27 Show that the c otangent lifte d action on S O ( n ) is expr esse d as Q · P = Q T P as matrix multiplic ation. Deﬁnition 18.28 A momentum map is said to b e e quivariant when it is e quivariant with r esp e ct to the given action on P and the c o adjoint action on g ∗ . That is, J ( g · p ) = Ad ∗ g − 1 J ( p ) for every g ∈ G , p ∈ P , wher e g · p denotes the action of g on the p oint p and wher e A d denotes the adjoint action. Exercise 18.29 Show that the momentum map derive d fr om the c otangent lift in Theorem 18.19 is e quivariant. Example 18.30 (Momentum map for symplectic represen tations) L et ( V , Ω) b e a symple ctic ve ctor sp ac e and let G b e a Lie gr oup acting line arly and symple ctic al ly on V . This action admits an e quivariant momentum map J : V → g given by J ξ ( v ) = h J ( v ) , ξ i = 1 2 Ω( ξ · v , v ) , wher e ξ · v denotes the Lie algebr a r epr esentation of the element ξ ∈ g on the ve ctor v ∈ V . T o verify this, note that the inﬁnitesimal gener ator ξ V ( v ) = ξ · v , by the deﬁnition of the Lie algebr a r epr esentation induc e d by the given Lie gr oup r epr esentation, and that Ω( ξ · u, v ) = − Ω( u, ξ · v ) for al l u, v ∈ V . Ther efor e d J ξ ( u )( v ) = 1 2 Ω( ξ · u, v ) + 1 2 Ω( ξ · v , u ) = Ω( ξ · u, v ) . Equivarianc e of J fol lows fr om the obvious r elation g − 1 · ξ · g · v = (Ad g − 1 ξ ) · v for any g ∈ G , ξ ∈ g , and v ∈ V . D. D. Holm Imp erial College London Applications of P oisson Geometry 80 Example 18.31 (Cayley-Klein parameters and the Hopf ﬁbration) Consider the natur al ac- tion of S U (2) on C 2 . Sinc e this action is by isometries of the Hermitian metric, it is automatic al ly symple ctic and ther efor e has a momentum map J : C 2 → su (2) ∗ given in example 18.30, that is, h J ( z , w ) , ξ i = 1 2 Ω( ξ · ( z , w ) , ( z , w )) , wher e z , w ∈ C and ξ ∈ su (2) . Now the symple ctic form on C 2 is given by minus the imaginary p art of the Hermitian inner pr o duct. That is, C n has Hermitian inner pr o duct given by z · w := P n j =1 z j w j , wher e z = ( z 1 , . . . , z n ) , w = ( w 1 , . . . , w n ) ∈ C n . The symple ctic form is thus given by Ω( z , w ) := − Im( z · w ) and it is identic al to the one given b efor e on R 2 n by identifying z = u + i v ∈ C n with ( u , v ) ∈ R 2 n and w = u 0 + i v 0 ∈ C n with ( u 0 , v 0 ) ∈ R 2 n . The Lie algebr a su (2) of S U (2) c onsists of 2 × 2 skew Hermitian matric es of tr ac e zer o. This Lie algebr a is isomorphic to so (3) and ther efor e to ( R 3 , × ) by the isomorphism given by x = ( x 1 , x 2 , x 3 ) ∈ R 3 7→ e x := 1 2  − ix 3 − ix 1 − x 2 − ix 1 + x 2 ix 3  ∈ su (2) . Thus we have [ e x , e y ] = ( x × y ) e for any x , y ∈ R 3 . Other useful r elations ar e det(2 e x ) = k x k 2 and trace( e x e y ) = − 1 2 x · y . Identify su (2) ∗ with R 3 by the map µ ∈ su (2) ∗ 7→ ˇ µ ∈ R 3 deﬁne d by ˇ µ · x := − 2 h µ, e x i for any x ∈ R 3 . With these notations, the momentum map ˇ J : C 2 → R 3 c an b e explicitly c ompute d in c o or dinates: for any x ∈ R 3 we have ˇ J ( z , w ) · x = − 2 h J ( z , w ) , e x i = 1 2 Im  − ix 3 − ix 1 − x 2 − ix 1 + x 2 ix 3   z w  ·  z w  = − 1 2 (2 Re( w z ) , 2 Im( w z ) , | z | 2 − | w | 2 ) · x . Ther efor e ˇ J ( z , w ) = − 1 2 (2 w z , | z | 2 − | w | 2 ) ∈ R 3 . Thus, ˇ J is a Poisson map fr om C 2 , endowe d with the c anonic al symple ctic structur e, to R 3 , endowe d with the + Lie Poisson structur e. Ther efor e, − ˇ J : C 2 → R 3 is a c anonic al map, if R 3 has the − Lie-Poisson br acket r elative to which the fr e e rigid b o dy e quations ar e Hamiltonian. Pul ling b ack the Hamiltonian H ( Π ) = Π · I − 1 Π / 2 to C 2 gives a Hamiltonian function (c al le d c ol le ctive) on C 2 . The classic al Hamilton e quations for this function ar e ther efor e pr oje cte d by − ˇ J to the rigid b o dy e quations ˙ Π = Π × I − 1 Π . In this c ontext, the variables ( z , w ) ar e c al le d the Cayley-Klein p ar ameters . Exercise 18.32 Show that − ˇ J | S 3 : S 3 → S 2 is the Hopf ﬁbr ation . In other wor ds, the momentum map of the S U (2) -action on C 2 , the Cayley-Klein p ar ameters and the family of Hopf ﬁbr ations on c onc entric thr e e-spher es in C 2 ar e al l the same map. Exercise 18.33 Optical trav eling wa ve pulses The e quation for the evolution of the c omplex amplitude of a p olarize d optic al tr aveling wave pulse in a material me dium is given as ˙ z i = 1 √ − 1 ∂ H ∂ z ∗ i D. D. Holm Imp erial College London Applications of P oisson Geometry 81 with Hamiltonian H : C 2 → R deﬁne d by H = z ∗ i χ (1) ij z j + 3 z ∗ i z ∗ j χ (3) ij kl z k z l and the c onstant c omplex tensor c o eﬃcients χ (1) ij and χ (1) ij kl have the pr op er Hermitian and p ermutation symmetries for H to b e r e al. Deﬁne the Stokes ve ctors by the isomorphism, u = ( u 1 , u 2 , u 3 ) ∈ R 3 7→ e u := 1 2  − iu 3 − iu 1 − u 2 − iu 1 + u 2 iu 3  ∈ su (2) . 1. Pr ove that this isomorphism is an e quivariant momentum map. 2. De duc e the e quations of motion for the Stokes ve ctors of this optic al tr aveling wave and write it as a Lie Poisson Hamiltonian system. 3. Determine how this system is r elate d to the e quations for an S O (3) rigid b o dy. Exercise 18.34 The formula determining the momentum map for the c otangent-lifte d action of a Lie gr oup G on a smo oth manifold Q may b e expr esse d in terms of the p airing h · , · i : g ∗ × g 7→ R as h J , ξ i = h p , £ ξ q i , wher e ( q , p ) ∈ T ∗ q Q and £ ξ q is the inﬁnitesimal gener ator of the action of the Lie algebr a element ξ on the c o or dinate q . Deﬁne appr opriate p airings and determine the momentum maps explicitly for the fol lowing actions, [a] £ ξ q = ξ × q for R 3 × R 3 7→ R 3 [b] £ ξ q = ad ξ q for ad-action ad : g × g 7→ g in a Lie algebr a g [c] Aq A − 1 for A ∈ GL (3 , R ) acting on q ∈ GL (3 , R ) by matrix c onjugation [d] Aq for left action of A ∈ S O (3) on q ∈ S O (3) [e] Aq A T for A ∈ GL (3 , R ) acting on q ∈ S y m (3) , that is q = q T . Answ er 18.35 [a] p · ξ × q = q × p · ξ ⇒ J = q × p . (The p airing is sc alar pr o duct of ve ctors.) [b] h p , ad ξ q i = − h ad ∗ q p , ξ i ⇒ J = ad ∗ q p for the p airing h · , · i : g ∗ × g 7→ R [c] Compute T e ( Aq A − 1 ) = ξ q − qξ = [ ξ , q ] for ξ = A 0 (0) ∈ g l (3 , R ) acting on q ∈ GL (3 , R ) by matrix Lie br acket [ · , · ] . F or the matrix p airing h A , B i = trace( A T B ) , we have trace( p T [ ξ , q ]) = trace(( pq T − q T p ) T ξ ) ⇒ J = pq T − q T p . [d] Compute T e ( Aq ) = ξ q for ξ = A 0 (0) ∈ so (3) acting on q ∈ S O (3) by left matrix multiplic ation. F or the matrix p airing h A , B i = trace( A T B ) , we have trace( p T ξ q ) = trace(( pq T ) T ξ ) ⇒ J = 1 2 ( pq T − q T p ) , wher e we have use d antisymmetry of the matrix ξ ∈ so (3) . [e] Compute T e ( Aq A T ) = ξ q + q ξ T for ξ = A 0 (0) ∈ g l (3 , R ) acting on q ∈ S y m (3) . F or the matrix p air- ing h A , B i = trace( A T B ) , we have trace( p T ( ξ q + q ξ T )) = trace( q ( p T + p ) ξ ) = trace(2 q p ) T ξ ) ⇒ J = 2 q p , wher e we have use d symmetry of the matrix ξ q + qξ T to cho ose p = p T . (The momen- tum c anonic al to the symmetric matrix q = q T should b e symmetric to have the c orr e ct numb er of c omp onents!) D. D. Holm Imp erial College London Applications of P oisson Geometry 82 Equiv ariance Deﬁnition 18.36 A momentum map is Ad ∗ - equiv ariant iﬀ J ( g · x ) = Ad ∗ g − 1 J ( x ) for al l g ∈ G, x ∈ P . Prop osition 18.37 Al l c otangent-lifte d actions ar e Ad ∗ -e quivariant. Prop osition 18.38 Every Ad ∗ -e quivariant momentum map J : P → g ∗ is a Poisson map, with r esp e ct to the ‘+’ Lie-Poisson br acket on g ∗ . 19 Quic k summary for momen tum maps Let G b e a Lie group, g its Lie algebra, and let g ∗ b e its dual. Supp ose that G acts symplectically on a symplectic manifold P with symplectic form denoted b y Ω. Denote the inﬁnitesimal generator asso ciated with the Lie algebra element ξ by ξ P and let the Hamiltonian vector ﬁeld asso ciated to a function f : P → R b e denoted X f , so that d f = X f Ω. 19.1 Deﬁnition, History and Ov erview A momentum map J : P → g ∗ is deﬁned by the condition relating the inﬁnitesimal generator ξ P of a symmetry to the v ector ﬁeld of its corresp onding conserv ation law, h J, ξ i , ξ P = X h J,ξ i for all ξ ∈ g . Here h J, ξ i : P → R is deﬁned by the natural point wise pairing. A momen tum map is said to be e quivariant when it is equiv ariant with respect to the given action on P and the coadjoint action on g ∗ . That is, J ( g · p ) = Ad ∗ g − 1 J ( p ) for every g ∈ G , p ∈ P , where g · p denotes the action of g on the p oint p and where Ad denotes the adjoin t action. According to [W e1983], [Lie1890] already knew that 1. An action of a Lie group G with Lie algebra g on a symplectic manifold P should be accompanied b y such an equiv ariant momen tum map J : P → g ∗ and 2. The orbits of this action are themselv es symplectic manifolds. The links with mechanics were dev elop ed in the work of Lagrange, Poisson, Jacobi and, later, No ether. In particular, No ether show ed that a momen tum map for the action of a group G that is a symmetry of the Hamiltonian for a giv en system is a c onservation law for that system. In mo dern form, the momen tum map and its equiv ariance were rediscov ered in [Ko1966] and [So1970] in the general symplectic case, and in [Sm1970] for the case of the lifted action from a D. D. Holm Imp erial College London Applications of P oisson Geometry 83 manifold Q to its cotangen t bundle P = T ∗ Q . In this case, the equiv ariant momentum map is given explicitly b y h J ( α q ) , ξ i = h α q , ξ Q ( q ) i , where α q ∈ T ∗ Q , ξ ∈ g , and where the angular brack ets denote the natural pairing on the appropriate spaces. See [MaRa1994] and [OrRa2004] for additional history and description of the momentum map and its prop erties. 20 Rigid b o dy equations on SO( n ) Recall from [Man1976] and [Ra1980] that the left inv arian t generalized rigid b o dy equations on SO( n ) ma y b e written as ˙ Q = Q Ω , ˙ M = M Ω − Ω M =: [ M , Ω] , (RBn) where Q ∈ SO( n ) denotes the conﬁguration space v ariable (the attitude of the bo dy), Ω = Q − 1 ˙ Q ∈ so ( n ) is the b ody angular velocity , and M := J (Ω) = D 2 Ω + Ω D 2 ∈ so ∗ ( n ) , is the bo dy angular momen tum. Here J : so ( n ) → so ( n ) ∗ is the symmetric (with resp ect to the ab o ve inner pro duct) positive deﬁnite operator deﬁned by J (Ω) = D 2 Ω + Ω D 2 , where D 2 is the square of the constant diagonal matrix D = diag { d 1 , d 2 , d 3 } satisfying d 2 i + d 2 j > 0 for all i 6 = j . F or n = 3 the elemen ts of d 2 i are related to the standard diagonal moment of inertia tensor I by I = diag { I 1 , I 2 , I 3 } , I 1 = d 2 2 + d 2 3 , I 2 = d 2 3 + d 2 1 , I 3 = d 2 1 + d 2 2 . The Euler equations for the S O ( n ) rigid b o dy ˙ M = [ M , Ω] are readily chec ked to b e the Euler- Lagrange equations on so ( n ) for the Lagrangian L ( Q, ˙ Q ) = l (Ω) = 1 2 h Ω , J (Ω) i , with Ω = Q T ˙ Q. The momen tum is found via the Legendre transformation to b e ∂ l ∂ Ω = J (Ω) = M , and the corresp onding Hamiltonian is H ( M ) = ∂ l ∂ Ω · Ω − l (Ω) = 1 2  M , J − 1 ( M )  . The quantit y M is the angular momentum in the b o dy frame. The corresp onding angular momentum in space, m = QM Q T , is conserv ed ˙ m = 0 . Indeed, conserv ation of spatial angular momen tum m implies Euler’s equations for the b ody angular momen tum M = Q T mQ = Ad ∗ Q m . D. D. Holm Imp erial College London Applications of P oisson Geometry 84 20.1 Implications of left in v ariance This Hamiltonian H ( M ) is inv ariant under the action of S O ( n ) from the left. The corresp onding conserv ed momentum map under this symmetry is kno wn from the previous lecture as J L : T ∗ S O ( n ) 7→ so ( n ) ∗ is J L ( Q, P ) = P Q T On the other hand, w e know (from Lectures 18 & 19) that the momen tum map for righ t action is J R : T ∗ S O ( n ) 7→ so ( n ) ∗ , J R ( Q, P ) = Q T P Hence M = Q T P = J R . Therefore, one computes H ( Q, P ) = H ( Q, Q · M ) = H (Id , M ) (b y left inv ariance) = H ( M ) = 1 2 h M , J − 1 ( M ) i = 1 2 h Q T P , J − 1 ( Q T P ) i Hence, w e may write the S O ( n ) rigid bo dy Hamiltonian as H ( Q, P ) = 1 2 h Q T P , Ω( Q, P ) i Consequen tly , the v ariational deriv atives of H ( Q, P ) = 1 2 h Q T P , Ω( Q, P ) i are δ H =  Q T δ P + δ Q T P , Ω( Q, P )  = tr( δ P T Q Ω) + tr( P T δ Q Ω) = tr( δ P T Q Ω) + tr( δQ Ω P T ) = tr( δ P T Q Ω) + tr( δQ T P Ω T ) = h δ P , Q Ω i − h δ Q , P Ω i where skew symmetry of Ω is used in the last step, i.e., Ω T = − Ω. Th us, Hamilton’s canonical equations tak e the form, ˙ Q = δ H δ P = Q Ω , ˙ P = − δ H δ Q = P Ω . (20.1) Equations (20.1) are the symmetric gener alize d rigid b o dy e quations , deriv ed earlier in [BlCr1997] and [BlBrCr1997] from the viewp oin t of optimal con trol. Combining them yields, Q − 1 ˙ Q = Ω = P − 1 ˙ P ⇐ ⇒ ( P Q T ) ˙ = 0 , in agreement with conserv ation of the momen tum map J L ( Q, P ) = P Q T corresp onding to symmetry of the Hamiltonian under left action of S O ( n ). This momen tum map is the angular momentum in space, whic h is related to the angular momentum in the b o dy b y P Q T = m = QM Q T . Th us, we recognize the canonical momentum as P = QM (see exercise 18.23), and the momentum maps for left and righ t actions as, J L = m = P Q T (spatial angular momen tum) J R = M = Q T P (b ody angular momentum) Th us, momentum maps T G ∗ 7→ g ∗ corresp onding to symmetries of the Hamiltonian pro duce conser- v ation laws; while momentum maps T G ∗ 7→ g ∗ whic h do not corresp ond to symmetries may b e used to re-express the equations on g ∗ , in terms of v ariables on T G ∗ . D. D. Holm Imp erial College London Applications of P oisson Geometry 85 21 Manak o v’s form ulation of the S O (4) rigid b o dy The Euler equations on S O (4) are dM dt = M Ω − Ω M = [ M , Ω] , (RBn) where Ω and M are skew symmetric 4 × 4 matrices. The angular frequency Ω is a linear function of the angular momen tum, M . [Man1976] “deformed” these equations into d dt ( M + λA ) = [( M + λA ) , (Ω + λB )] , where A , B are also skew symmetric 4 × 4 matrices and λ is a scalar constant parameter. F or these equations to hold for an y v alue of λ , the co eﬃcen t of eac h p o wer m ust v anish. • The co eﬃcen t of λ 2 is 0 = [ A, B ] So A and B m ust comm ute. So, let them b e constan t and diagonal: A ij = diag( a i ) δ ij , B ij = diag( b i ) δ ij (no sum) • The co eﬃcen t of λ is 0 = dA dt = [ A, Ω] + [ M , B ] Therefore, b y antisymmetry of M and Ω, ( a i − a j )Ω ij = ( b i − b j ) M ij ⇐ ⇒ Ω ij = b i − b j a i − a j M ij (no sum) • Finally , the co eﬃcen t of λ 0 is the Euler equation, dM dt = [ M , Ω] , but no w with the restriction that the moments of inertia are of the form, Ω ij = b i − b j a i − a j M ij (no sum) whic h turns out to p ossess only 5 free parameters. With these conditions, Manako v’s deformation of the S O (4) rigid b ody implies for every p o w er n that d dt ( M + λA ) n = [( M + λA ) n , (Ω + λB )] , Since the comm utator is antisymmetric, its trace v anishes and one has d dt trace( M + λA ) n = 0 after comm uting the trace op eration with time deriv ative. Consequen tly , trace( M + λA ) n = constan t for each p ow er of λ . That is, all the co eﬃcien ts of each p o wer of λ are constant in time for the S O (4) rigid bo dy . [Man1976] pro ved that these constan ts of motion are suﬃcien t to completely determine the solution. D. D. Holm Imp erial College London Applications of P oisson Geometry 86 Remark 21.1 This r esult gener alizes c onsider ably. First, it holds for S O ( n ) . Inde e d, as as pr oven using the the ory of algebr aic varieties in [Ha1984], Manakov’s metho d c aptur es al l the algebr aic al ly inte gr able rigid b o dies on S O ( n ) and the moments of inertia of these b o dies p ossess only 2 n − 3 p ar am- eters. (R e c al l that in Manakov’s c ase for S O (4) the moment of inertia p ossesses only ﬁve p ar ameters.) Mor e over, [MiF o1978] pr ove that every c omp act Lie gr oup admits a family of left-invariant metrics with c ompletely inte gr able ge o desic ﬂows. Exercise 21.2 T ry c omputing the c onstants of motion trace( M + λA ) n for the values n = 2 , 3 , 4 . How many additional c onstants of motion ar e ne e de d for inte gr ability for these c ases? How many for gener al n ? Hint: ke ep in mind that M is a skew symmetric matrix, M T = − M , so the tr ac e of the pr o duct of any diagonal matrix times an o dd p ower of M vanishes. Answ er 21.3 The tr ac es of the p owers trace( M + λA ) n ar e given by n=2 : tr M 2 + 2 λ tr ( AM ) + λ 2 tr A 2 n=3 : tr M 3 + 3 λ tr ( AM 2 ) + 3 λ 2 tr A 2 M + λ 3 tr A 3 n=4 : tr M 4 + 4 λ tr ( AM 3 ) + λ 2 (2tr A 2 M 2 + 4tr AM AM ) + λ 3 tr A 3 M + λ 4 tr A 4 The numb er of c onserve d quantities for n = 2 , 3 , 4 ar e, r esp e ctively, one ( C 1 = tr M 2 ), one ( I 1 = tr AM 2 ) and two ( C 2 = tr M 4 and I 2 = 2tr A 2 M 2 + 4tr AM AM ). The quantities C 1 and C 2 ar e Casimirs for the Lie-Poisson br acket for the rigid b o dy. Thus, { C 1 , H } = 0 = { C 2 , H } for any Hamiltonian H ( M ) ; so of c ourse C 1 and C 2 ar e c onserve d. However, e ach Casimir only r e duc es the dimension of the system by one. The dimension of the original phase sp ac e is dim T ∗ S O ( n ) = n ( n − 1) . This is r e duc e d in half by left invarianc e of the Hamiltonian to the dimension of the dual Lie algebr a dim so ( n ) ∗ = n ( n − 1) / 2 . F or n = 4 , dim so (4) ∗ = 6 . One then subtr acts the numb er of Casimirs (two) by p assing to their level surfac es, which le aves four dimensions r emaining in this c ase. The other two c onstants of motion I 1 and I 2 turn out to b e suﬃcient for inte gr ability, b e c ause they ar e in involution { I 1 , I 2 } = 0 and b e c ause the level surfac es of the Casimirs ar e symple ctic manifolds, by the Marsden-Weinstein r e duction the or em [MaWe74]. F or mor e details, se e [R a1980]. Exercise 21.4 How do the Euler e quations lo ok on so (4) ∗ as a matrix e quation? Is ther e an analo g of the hat map for so (3) ∗ ? Hint: the Lie algebr a so (4) is lo c al ly isomorphic to so (3) × so (3) . Exercise 21.5 Write Manakov’s deformation of the rigid b o dy e quations in the symmetric form (20.1). 22 F ree ellipsoidal motion on GL( n ) Riemann [Ri1860] considered the deformation of a b ody in R n giv en by x ( t, x 0 ) = Q ( t ) x 0 , (22.1) with x , x 0 ∈ R n , Q ( t ) ∈ GL + ( n, R ) and x ( t 0 , x 0 ) = x 0 , so that Q ( t 0 ) = I d . (The subscript + in GL + ( n, R ) means n × n matrices with positive determinan t.) Thus, x ( t, x 0 ) is the curren t (Eulerian) p osition at time t of a material parcel that was at (Lagrangian) p osition x 0 at time t 0 . The “defor- mation gradient,” that is, the Jacobian matrix Q = ∂ x/∂ x 0 of this “Lagrange-to-Euler map,” is a function of only time, t , ∂ x/∂ x 0 = Q ( t ) , with det Q > 0 . D. D. Holm Imp erial College London Applications of P oisson Geometry 87 The v elo cit y of such a motion is giv en b y ˙ x ( t, x 0 ) = ˙ Q ( t ) x 0 = ˙ Q ( t ) Q − 1 ( t ) x = u ( t, x ) . (22.2) The kinetic energy for suc h a b o dy occupying a reference v olume B deﬁnes the quadratic form, L = 1 2 Z B ρ ( x 0 ) | ˙ x ( t, x 0 ) | 2 d 3 x 0 = 1 2 tr  ˙ Q ( t ) T I ˙ Q ( t )  = 1 2 ˙ Q i A I AB ˙ Q i B . Here I is the constant symmetric tensor, I AB = Z B ρ ( x 0 ) x A 0 x B 0 d 3 x 0 , whic h we will take as b eing prop ortional to the identit y I AB = c 2 0 δ AB for the remainder of these considerations. This corresp onds to taking an initially spherical reference conﬁguration for the ﬂuid. Hence, w e are dealing with the Lagrangian consisting only of kinetic energy , 5 L = 1 2 tr  ˙ Q ( t ) T ˙ Q ( t )  . The Euler-Lagrange equations for this Lagrangian simply represen t fr e e motion on the group GL + ( n, R ), ¨ Q ( t ) = 0 , whic h is immediately integrable as Q ( t ) = Q (0) + ˙ Q (0) t , where Q (0) and ˙ Q (0) are the v alues at the initial time t = 0. Legendre transforming this Lagrangian for free motion yields P = ∂ L ∂ ˙ Q T = ˙ Q . The corresp onding Hamiltonian is expressed as H ( Q, P ) = 1 2 tr  P T P  = 1 2 k P k 2 . The canonical equations for this Hamiltonian are simply ˙ Q = P , with ˙ P = 0 . 22.1 P olar decomp osition of free motion on GL + ( n, R ) The deformation tensor Q ( t ) ∈ GL + ( n, R ) for such a bo dy ma y b e decomp osed as Q ( t ) = R − 1 ( t ) D ( t ) S ( t ) . (22.3) This is the p olar decomp osition of a matrix in GL + ( n, R ). The in terpretations of the v arious comp o- nen ts of the motion can b e seen from equation (22.1). Namely , • R ∈ S O ( n ) rotates the x -co ordinates, 5 [Ri1860] considered the muc h more diﬃcult problem of a self-gr avitating ellipsoid deforming according to (22.1) in R 3 . See [Ch1969] for the history of this problem. D. D. Holm Imp erial College London Applications of P oisson Geometry 88 • S ∈ S O ( n ) rotates the x 0 -co ordinates in the reference conﬁguration 6 and • D is a diagonal matrix whic h represen ts stretching deformations along the principal axes of the b ody . The t wo S O ( n ) rotations lead to their corresponding angular frequencies, deﬁned by Ω = ˙ RR − 1 , Λ = ˙ S S − 1 . (22.4) Rigid b o dy motion will result, when S restricts to the iden tity matrix and D is a constant diagonal matrix. Remark 22.1 The c ombine d motion of a set of ﬂuid p ar c els governe d by (22.1) along the curve Q ( t ) ∈ GL + ( n, R ) is c al le d “el lipsoidal,” b e c ause it c an b e envisione d in thr e e dimensions as a ﬂuid el lipsoid whose orientation in sp ac e is governe d by R ∈ S O ( n ) , whose shap e is determine d by D c onsisting of its instantane ous principle axes lengths and whose internal cir culation of material is describ e d by S ∈ S O ( n ) . In addition, ﬂuid p ar c els initial ly arr ange d along a str aight line within the el lipse wil l r emain on a str aight line. 22.2 Euler-P oincar´ e dynamics of free Riemann ellipsoids In Hamilton’s principle, δ R L dt = 0, w e chose a Lagrangian L : T GL + ( n, R ) → R in the form L ( Q, ˙ Q ) = T (Ω , Λ , D , ˙ D ) , (22.5) in which the kinetic energy T is given b y using the p olar decomposition Q ( t ) = R − 1 ( t ) D ( t ) S ( t ) in (22.3), as follo ws. ˙ Q = R − 1 ( − Ω D + ˙ D + D Λ) S . (22.6) Consequen tly , the kinetic energy for ellipsoidal motion b ecomes T = 1 2 trace h − Ω D 2 Ω − Ω D ˙ D + Ω D Λ D + ˙ D D Ω + ˙ D 2 − D Λ 2 D − ˙ D Λ D + D Λ D Ω + D Λ ˙ D i = 1 2 trace h − Ω 2 D 2 − Λ 2 D 2 + 2Ω D Λ D | {z } Coriolis coupling + ˙ D 2 i . (22.7) Remark 22.2 Note the discr ete exchange symmetry of the kinetic ener gy: T is invariant under Ω ↔ Λ . 7 F or Λ = 0 and D constant expression (22.7) for T reduces to the usual kinetic energy for the rigid-b ody , T    Λ=0 , D = const = − 1 4 trace h Ω( D Ω + Ω D ) i . (22.8) 6 This is the “particle relab eling map” for this class of motions. 7 According to [Ch1969] this discrete symmetry was ﬁrst noticed by Riemann’s friend, [De1860]. D. D. Holm Imp erial College London Applications of P oisson Geometry 89 This Lagrangian (22.5) is inv ariant under the right action, R → R g and S → S g , for g ∈ S O ( n ). In taking v ariations we shall use the formulas 8 δ Ω = ˙ Σ + [Σ , Ω] ≡ ˙ Σ − ad Ω Σ , Σ ≡ δ R R − 1 , (22.9) δ Λ = ˙ Ξ + [Ξ , Λ] ≡ ˙ Ξ − ad Λ Ξ , Ξ ≡ δ S S − 1 , (22.10) in which the ad-op eration is deﬁned in terms of the Lie-algebra (matrix) comm utator [ · , · ] as, e.g., ad Ω Σ ≡ [Ω , Σ ]. Substituting these form ulas into Hamilton’s principle giv es 0 = δ Z L dt = Z dt ∂ L ∂ Ω · δ Ω + ∂ L ∂ Λ · δ Λ + ∂ L ∂ D δ D + ∂ L ∂ ˙ D δ ˙ D , = Z dt ∂ L ∂ Ω · h ˙ Σ − ad Ω Σ i + ∂ L ∂ Λ · h ˙ Ξ − ad Λ Ξ i + h ∂ L ∂ D − d dt ∂ L ∂ ˙ D i δ D , = − Z dt h d dt ∂ L ∂ Ω − ad ∗ Ω δ L δ Ω i · Σ + h d dt ∂ L ∂ Λ − ad ∗ Λ ∂ L ∂ Λ i · Ξ (22.11) + h d dt ∂ L ∂ ˙ D − ∂ L ∂ D i δ D , where, the op eration ad ∗ Ω , for example, is deﬁned b y ad ∗ Ω ∂ L ∂ Ω · Σ = − ∂ L ∂ Ω · ad Ω Σ = − ∂ L ∂ Ω · [Ω , Σ ] , (22.12) and the dot ‘ · ’ denotes pairing b et w een the Lie algebra and its dual. This could also hav e b een written in the notation using h· , ·i : g ∗ × g → R as,  ad ∗ Ω ∂ L ∂ Ω , Σ  = −  ∂ L ∂ Ω , ad Ω Σ  = −  ∂ L ∂ Ω , [Ω , Σ ]  . (22.13) The Euler-P oincar´ e dynamics is given b y the stationarity conditions for Hamilton’s principle, Σ : d dt ∂ L ∂ Ω − ad ∗ Ω ∂ L ∂ Ω = 0 , (22.14) Ξ : d dt ∂ L ∂ Λ − ad ∗ Λ ∂ L ∂ Λ = 0 , (22.15) δ D : d dt ∂ L ∂ ˙ D − ∂ L ∂ D = 0 . (22.16) These are the Euler-Poinc ar ´ e e quations for the ellipsoidal motions generated b y Lagrangians of the form given in equation (22.5). F or example, suc h Lagrangians determine the dynamics of the Riemann ellipsoids – circulating, rotating, self-gravitating ﬂuid ﬂows at constant densit y within an ellipsoidal b oundary . 22.3 Left and righ t momentum maps: Angular momen tum versus circulation The Euler-P oincar´ e equations (22.14-22.16) in volv e angular momenta deﬁned in terms of the angular v elo cities Ω, Λ and the shap e D by M = ∂ T ∂ Ω = − Ω D 2 − D 2 Ω + 2 D Λ D , (22.17) N = ∂ T ∂ Λ = − Λ D 2 − D 2 Λ + 2 D Ω D . (22.18) 8 These v ariational form ulas are obtained directly from the deﬁnitions of Ω and Λ. D. D. Holm Imp erial College London Applications of P oisson Geometry 90 These angular momenta are related to the original deformation gradient Q = R − 1 D S in equation (22.1) b y the tw o momen tum maps from Example 18.26 P Q T − QP T = ˙ QQ T − Q ˙ Q T = R − 1 M R , (22.19) P T Q − Q T P = ˙ Q T Q − Q T ˙ Q = S − 1 N S . (22.20) T o see that N is related to the vorticity , w e consider the exterior deriv ative of the circulation one-form u · d x deﬁned as d ( u · d x ) = curl u · d S = 1 2 ( ˙ Q T Q − Q T ˙ Q ) j k dx j 0 ∧ dx k 0 = ( S − 1 N S ) j k dx j 0 ∧ dx k 0 . (22.21) Th us, S − 1 N S is the ﬂuid vorticit y , referred to the Lagrangian co ordinate frame. F or Euler’s ﬂuid equations, Kelvin’s circulation theorem implies ( S − 1 N S ) ˙ = 0. Lik ewise, M is related to the angular momentum by considering u i x j − u j x i = ˙ Q ik x k 0 x l 0 Q T lj − Q ik x k 0 x l 0 ˙ Q T lj . (22.22) F or spherical symmetry , w e may choose x k 0 x l 0 = δ kl and, in this case, the previous expression b ecomes u i x j − u j x i = [ ˙ QQ T − Q ˙ Q T ] ij = [ R − 1 M R ] ij . (22.23) Th us, R − 1 M R is the angular momentum of the motion, referred to the Lagrangian co ordinate frame for spherical symmetry . In this case, the angular momen tum is conserved, so that ( R − 1 M R ) ˙ = 0. In terms of these angular momen ta, the Euler-Poincar ´ e-Lagrange equations (22.14-22.16) are ex- pressed as ˙ M = [Ω , M ] , (22.24) ˙ N = [Λ , N ] , (22.25) d dt  ∂ L ∂ ˙ D  = ∂ L ∂ D . (22.26) P erhaps not unexpectedly , b ecause of the combined symmetries of the kinetic-energy Lagrangian (22.5) under both left and righ t actions of S O ( n ), the ﬁrst t w o equations are consisten t with the conserv ation la ws, ( R − 1 M R ) ˙ = 0 and ( S − 1 N S ) ˙ = 0 , resp ectiv ely . Thus, equation (22.24) is the angular momentum equation while (22.25) is the vorticit y equation. (Fluids ha ve b oth types of circulatory motions.) The remaining equation (22.26) for the diagonal matrix D determines the shap e of the ellipsoid undergoing free motion on GL ( n, R ). 22.4 V ector represen tation of free Riemann ellipsoids in 3D In three dimensions these expressions may b e written in v ector form b y using the hat map , written no w using upp er and lo wer case Greek letters as, Ω ij =  ij k ω k , Λ ij =  ij k λ k , with  123 = 1, and D = diag { d 1 , d 2 , d 3 } . Exercise 22.3 What is the analo g of the hat map in four dimensions? Hint: lo c al ly the Lie algebr a so (4) is isomorphic to so (3) × so (3) . D. D. Holm Imp erial College London Applications of P oisson Geometry 91 Hence, the angular-motion terms in the kinetic energy ma y b e rewritten as − 1 2 trace (Ω 2 D 2 ) = 1 2 h ( d 2 1 + d 2 2 ) ω 2 3 + ( d 2 2 + d 2 3 ) ω 2 1 + ( d 2 3 + d 2 1 ) ω 2 2 i , (22.27) − 1 2 trace (Λ 2 D 2 ) = 1 2 h ( d 2 1 + d 2 2 ) λ 2 3 + ( d 2 2 + d 2 3 ) λ 2 1 + ( d 2 3 + d 2 1 ) λ 2 2 i , (22.28) and − 1 2 trace (Ω D Λ D ) = h d 1 d 2 ( ω 3 λ 3 ) + d 2 d 3 ( ω 1 λ 1 ) + d 3 d 1 ( ω 2 λ 2 ) i . (22.29) On comparing equations (22.8) and (22.27) for the kinetic energy of the rigid b o dy part of the motion, w e identify the usual momen ts of inertia as I k = d 2 i + d 2 j , with i, j, k cyclic. The an tisymmetric matrices M and N hav e v ector representations in 3D giv en b y M k = ∂ T ∂ ω k = ( d 2 i + d 2 j ) ω k − 2 d i d j λ k , (22.30) N k = ∂ T ∂ λ k = ( d 2 i + d 2 j ) λ k − 2 d i d j ω k , (22.31) again with i , j , k cyclic p ermutations of { 1 , 2 , 3 } . V ector represen tation in 3D In terms of their 3D vector represen tations of the angular momenta in equations (22.30) and (22.31), the t wo equations (22.24) and (22.25) become ˙ M = ( ˙ RR − 1 ) M = Ω M = ω × M , ˙ N = ( ˙ S S − 1 ) N = Λ N = λ × N . (22.32) Relativ e to the Lagrangian ﬂuid frame of reference, these equations b ecome ( R − 1 M ) ˙ = R − 1 ( ˙ M − ω × M ) = 0 , (22.33) ( S − 1 N ) ˙ = S − 1 ( ˙ N − λ × N ) = 0 . (22.34) So each of these degrees of freedom represents a rotating, deforming b o dy , whose ellipsoidal shap e is go verned b y the Euler-Lagrange equations (22.26) for the lengths of its three principal axes. Exercise 22.4 (Elliptical motions with p oten tial energy on GL (2 , R ) ) Compute e quations (22.24- 22.26) for el liptic al motion in t he plane. Find what p otentials V ( D ) ar e solvable for L = T (Ω , Λ , D , ˙ D ) − V ( D ) by r e ducing these e quations to the sep ar ate d Newtonian forms, d 2 r 2 dt 2 = − dV ( r ) dr 2 , d 2 α dt 2 = − dW ( α ) dα , for r 2 = d 2 1 + d 2 2 and α = tan − 1 ( d 2 /d 1 ) with d 1 ( t ) and d 2 ( t ) in two dimensions. Hint: c onsider the p otential ener gy, V ( D ) = V  tr D 2 , det( D )  , for which the e quations b e c ome homo gene ous in r 2 ( t ) . D. D. Holm Imp erial College London Applications of P oisson Geometry 92 Exercise 22.5 (Ellipsoidal motions with p oten tial energy on GL (3 , R ) ) Cho ose the L agr angian in 3D, L = 1 2 tr  ˙ Q T ˙ Q  − V  tr ( Q T Q ) , det( Q )  , wher e Q ( t ) ∈ GL (3 , R ) is a 3 × 3 matrix function of time and the p otential ener gy V is an arbitr ary function of tr ( Q T Q ) and det( Q ) . 1. L e gendr e tr ansform this L agr angian. That is, ﬁnd the momenta P ij c anonic al ly c onjugate to Q ij , c onstruct the Hamiltonian H ( Q, P ) and write Hamilton ’s c anonic al e quations of motion for this pr oblem. 2. Show that the Hamiltonian is invariant under Q → O Q wher e O ∈ S O (3) . Construct the c otangent lift of this action on P . Henc e, c onstruct the momentum map of this action. 3. Construct another distinct action of S O (3) on this system which also le aves its Hamiltonian H ( Q, P ) invariant. Construct its momentum map. Do the two momentum maps Poisson c om- mute? Why? 4. How ar e these two momentum maps r elate d to the angular momentum and cir culation in e qua- tions (22.17) and (22.18)? 5. How do es the 2D r estriction of this pr oblem inform the pr evious one? Exercise 22.6 ( GL ( n, R ) − in v arian t motions) Be gin with the L agr angian L = 1 2 tr  ˙ S S − 1 ˙ S S − 1  + 1 2 ˙ q T S − 1 ˙ q wher e S is an n × n symmetric matrix and q ∈ R n is an n − c omp onent c olumn ve ctor. 1. L e gendr e tr ansform to c onstruct the c orr esp onding Hamiltonian and c anonic al e quations. 2. Show that the system is invariant under the gr oup action q → A q and S → AS A T for any c onstant invertible n × n matrix, A . 3. Compute the inﬁnitesimal gener ator for this gr oup action and c onstruct its c orr esp onding mo- mentum map. Is this momentum map e quivariant? 4. V erify dir e ctly that this momentum map is a c onserve d n × n matrix quantity by using the e quations of motion. 5. Is this system c ompletely inte gr able for any value of n > 2 ? 23 Hea vy top equations 23.1 In tro duction and deﬁnitions A top is a rigid b o dy of mass m rotating with a ﬁxed point of supp ort in a constan t gravitational ﬁeld of acceleration − g ˆ z pointing v ertically down ward. The orien tation of the b o dy relativ e to the v ertical axis ˆ z is deﬁned b y the unit vector Γ = R − 1 ( t ) ˆ z for a curve R ( t ) ∈ S O (3). According to its D. D. Holm Imp erial College London Applications of P oisson Geometry 93 deﬁnition, the unit vector Γ represents the motion of the vertical direction as seen from the rotating b ody . Consequently , it satisﬁes the auxiliary motion equation, ˙ Γ = − R − 1 ˙ R ( t ) Γ = Γ × Ω . Here the rotation matrix R ( t ) ∈ S O (3), the sk ew matrix ˆ Ω = R − 1 ˙ R ∈ so (3) and the bo dy angular frequency v ector Ω ∈ R 3 are related by the hat map, Ω =  R − 1 ˙ R  ˆ , where ˆ : ( so (3) , [ · , · ]) → ( R 3 , × ) with ˆ Ωv = Ω × v for any v ∈ R 3 . The motion of a top is determined from Euler’s equations in v ector form, I ˙ Ω = I Ω × Ω + mg Γ × χ , (23.1) ˙ Γ = Γ × Ω , (23.2) where Ω , Γ , χ ∈ R 3 are v ectors in the rotating b o dy frame. Here • Ω = (Ω 1 , Ω 2 , Ω 3 ) is the b ody angular velocity v ector. • I = diag ( I 1 , I 2 , I 3 ) is the momen t of inertia tensor, diagonalized in the b o dy principle axes. • Γ = R − 1 ( t ) ˆ z represen ts the motion of the unit vector along the v ertical axis, as seen from the b ody . • χ is the constan t vector in the bo dy from the p oin t of supp ort to the bo dy’s center of mass. • m is the total mass of the b ody and g is the constan t acceleration of gra vity . 23.2 Hea vy top action principle Prop osition 23.1 The he avy top e quations ar e e quivalent to the he avy top action principle for a r e duc e d action δ S red = 0 , with S red = Z b a l ( Ω , Γ ) dt = Z b a 1 2 h I Ω , Ω i − h mg χ , Γ i dt, (23.3) wher e variations of Ω and Γ ar e r estricte d to b e of the form δ Ω = ˙ Σ + Ω × Σ and δ Γ = Γ × Σ , (23.4) arising fr om variations of the deﬁnitions Ω =  R − 1 ˙ R  ˆ and Γ = R − 1 ( t ) ˆ z in which Σ ( t ) =  R − 1 δ R  ˆ is a curve in R 3 that vanishes at the endp oints in time. Pro of. Since I is symmetric and χ is constant, we obtain the v ariation, δ Z b a l ( Ω , Γ ) dt = Z b a h I Ω , δ Ω i − h mg χ , δ Γ i dt = Z b a h I Ω , ˙ Σ + Ω × Σ i − h mg χ , Γ × Σ i dt = Z b a  − d dt I Ω , Σ  + h I Ω , Ω × Σ i − h mg χ , Γ × Σ i dt = Z b a  − d dt I Ω + I Ω × Ω + mg Γ × χ , Σ  dt, D. D. Holm Imp erial College London Applications of P oisson Geometry 94 up on in tegrating by parts and using the endp oin t conditions, Σ ( b ) = Σ ( a ) = 0. Since Σ is otherwise arbitrary , (23.3) is equiv alent to − d dt I Ω + I Ω × Ω + mg Γ × χ = 0 , whic h is Euler’s motion equation for the heavy top (23.1). This motion equation is completed by the auxiliary equation ˙ Γ = Γ × Ω in (23.2) arising from the deﬁnition of Γ . The Legendre transformation for l ( Ω , Γ ) giv es the b o dy angular momen tum Π = ∂ l ∂ Ω = I Ω . The w ell known energy Hamiltonian for the hea vy top then emerges as h ( Π , Γ ) = Π · Ω − l ( Ω , Γ ) = 1 2 h Π , I − 1 Π i + h mg χ , Γ i , (23.5) whic h is the sum of the kinetic and p oten tial energies of the top. The Lie-Poisson Equations. Let f , h : g ∗ → R b e real-v alued functions on the dual space g ∗ . Denoting elements of g ∗ b y µ , the functional deriv ative of f at µ is deﬁned as the unique elemen t δ f /δ µ of g deﬁned b y lim ε → 0 1 ε [ f ( µ + εδ µ ) − f ( µ )] =  δ µ, δ f δ µ  , (23.6) for all δ µ ∈ g ∗ , where h· , ·i denotes the pairing b et ween g ∗ and g . Deﬁnition 23.2 (Lie-Poisson brac kets & Lie-P oisson equations) The ( ± ) Lie-Poisson br ackets ar e deﬁne d by { f , h } ± ( µ ) = ±  µ,  δ f δ µ , δ h δ µ  = ∓  µ, ad δ h/δµ δ f δ µ  . (23.7) The corresp onding Lie-Poisson e quations , determined by ˙ f = { f , h } read ˙ µ = { µ, h } = ∓ ad ∗ δ h/δµ µ , (23.8) where one deﬁnes the ad ∗ op eration in terms of the pairing h· , ·i , b y { f , h } =  µ, ad δ h/δµ δ f δ µ  =  ad ∗ δ h/δµ µ, δ f δ µ  . The Lie-P oisson setting of mec hanics is a sp ecial case of the general theory of systems on P oisson manifolds, for which there is no w an extensive theoretical developmen t. (See Marsden and Ratiu [2003] for a start on this literature.) D. D. Holm Imp erial College London Applications of P oisson Geometry 95 23.3 Lie-P oisson brack ets and momen tum maps. An imp ortan t feature of the rigid b o dy brack et carries o ver to general Lie algebras. Namely , Lie- Poisson br ackets on g ∗ arise fr om c anonic al br ackets on the c otangent bund le (phase space) T ∗ G asso ciated with a Lie group G whic h has g as its asso ciated Lie algebra. Thus, the pro cess b y which the Lie-P oisson brack ets arise is the momen tum map T ∗ G 7→ g ∗ . F or example, a rigid bo dy is free to rotate ab out its center of mass and G is the (prop er) rotation group SO(3). The choice of T ∗ G as the primitive phase space is made according to the classical pro cedures of mechanics describ ed earlier. F or the description using Lagrangian mechanics, one forms the velocity phase space T G . The Hamiltonian description on T ∗ G is then obtained by standard pro cedures: Legendre transforms, etc. The passage from T ∗ G to the space of Π ’s (b o dy angular momen tum space) is determined by left translation on the group. This mapping is an example of a momentum map ; that is, a mapping whose comp onen ts are the “No ether quan tities” asso ciated with a symmetry group. The map from T ∗ G to g ∗ b eing a P oisson map is a gener al fact ab out momentum maps . The Hamiltonian point of view of all this is a standard sub ject. Remark 23.3 (Lie-Poisson description of the hea vy top) As it turns out, the underlying Lie algebr a for the Lie-Poisson description of the he avy top c onsists of the Lie algebr a se (3 , R ) of in- ﬁnitesimal Euclide an motions in R 3 . This is a bit surprising, b e c ause he avy top motion itself do es not actual ly arise thr ough actions of the Euclide an gr oup of r otations and tr anslations on the b o dy, sinc e the b o dy has a ﬁxe d p oint! Inste ad, the Lie algebr a se (3 , R ) arises for another r e ason asso ciate d with the br e aking of the SO(3) isotr opy by the pr esenc e of the gr avitational ﬁeld. This symmetry br e aking intr o duc es a semidir e ct-pr o duct Lie-Poisson structur e which happ ens to c oincide with the dual of the Lie algebr a se (3 , R ) in the c ase of the he avy top. As we shal l se e later, a close p ar al lel exists b etwe en this c ase and the Lie-Poisson structur e for c ompr essible ﬂuids. 23.4 The heavy top Lie-P oisson brac k ets The Lie algebr a of the sp e cial Euclide an gr oup in 3D is se (3) = R 3 × R 3 with the Lie brac ket [( ξ , u ) , ( η , v )] = ( ξ × η , ξ × v − η × u ) . (23.9) W e identify the dual space with pairs ( Π , Γ ); the corresp onding ( − ) Lie-Poisson brack et called the he avy top br acket is { f , h } ( Π , Γ ) = − Π · ∇ Π f × ∇ Π h − Γ ·  ∇ Π f × ∇ Γ h − ∇ Π h × ∇ Γ f  . (23.10) This Lie-Poisson brack et and the Hamiltonian (23.5) reco ver the equations (23.1) and (23.2) for the hea vy top, as ˙ Π = { Π , h } = Π × ∇ Π h + Γ × ∇ Γ h = Π × I − 1 Π + Γ × mg χ , ˙ Γ = { Γ , h } = Γ × ∇ Π h = Γ × I − 1 Π . Remark 23.4 (Semidirect pro ducts and symmetry breaking) The Lie algebr a of the Euclide an gr oup has a structur e which is a sp e cial c ase of what is c al le d a semidir e ct pr o duct . Her e, it is the D. D. Holm Imp erial College London Applications of P oisson Geometry 96 semidir e ct pr o duct action so (3) s R 3 of the Lie algebr a of r otations so (3) acting on the inﬁnitesimal tr anslations R 3 , which happ ens to c oincide with se (3 , R ) . In gener al, the Lie br acket for semidir e ct pr o duct action g s V of a Lie algebr a g on a ve ctor sp ac e V is given by h ( X , a ) , ( X , a ) i =  [ X , X ] , X ( a ) − X ( a )  in which X, X ∈ g and a, a ∈ V . Her e, the action of the Lie algebr a on the ve ctor sp ac e is denote d, e.g., X ( a ) . Usual ly, this action would b e the Lie derivative. Lie-Poisson br ackets deﬁne d on the dual sp ac es of semidir e ct pr o duct Lie algebr as tend to o c cur under r ather gener al cir cumstanc es when the symmetry in T ∗ G is br oken, e.g., r e duc e d to an isotr opy sub gr oup of a set of p ar ameters. In p articular, ther e ar e similarities in structur e b etwe en the Poisson br acket for c ompr essible ﬂow and that for the he avy top. In the latter c ase, the vertic al dir e ction of gr avity br e aks isotr opy of R 3 fr om S O (3) to S O (2) . The gener al the ory for semidir e ct pr o ducts is r eviewe d in a variety of plac es, including [MaR aWe84a, MaR aWe84b]. Many inter esting examples of Lie-Poisson br ackets on semidir e ct pr o ducts exist for ﬂuid dynamics. These semidir e ct-pr o duct Lie- Poisson Hamiltonian the ories r ange fr om simple ﬂuids, to char ge d ﬂuid plasmas, to magnetize d ﬂuids, to multiphase ﬂuids, to sup er ﬂuids, to Y ang-Mil ls ﬂuids, r elativistic, or not, and to liquid crystals. Se e, for example, [GiHoKu1982], [HoKu1982], [HoKu1983], [HoKu1988]. F or discussions of many of these the ories fr om the Euler-Poinc ar´ e viewp oint, se e [HoMaR a1998a] and [Ho2002a]. 23.5 The heavy top form ulation b y the Kaluza-Klein construction The Lagrangian in the heavy top action principle (23.3) ma y be transformed into a quadratic form. This is accomplished b y susp ending the system in a higher dimensional space via the Kaluza-Klein c onstruction . This construction pro ceeds for the heavy top as a sligh t modiﬁcation of the w ell-kno wn Kaluza-Klein construction for a c harged particle in a prescrib ed magnetic ﬁeld. Let Q K K b e the manifold S O (3) × R 3 with v ariables ( R , q ). On Q K K in tro duce the Kaluza-Klein L agr angian L K K : T Q K K ' T S O (3) × T R 3 7→ R as L K K ( R , q , ˙ R , ˙ q ; ˆ z ) = L K K ( Ω , Γ , q , ˙ q ) = 1 2 h I Ω , Ω i + 1 2 | Γ + ˙ q | 2 , (23.11) with Ω =  R − 1 ˙ R  ˆ and Γ = R − 1 ˆ z . The Lagrangian L K K is p ositiv e deﬁnite in ( Ω , Γ , ˙ q ); so it may b e regarded as the kinetic energy of a metric, the Kaluza-Klein metric on T Q K K . The Legendre transformation for L K K giv es the momenta Π = I Ω and p = Γ + ˙ q . (23.12) Since L K K do es not depend on q , the Euler-Lagrange equation d dt ∂ L K K ∂ ˙ q = ∂ L K K ∂ q = 0 , sho ws that p = ∂ L K K /∂ ˙ q is conserv ed. The c onstant ve ctor p is no w identiﬁed as the v ector in the b ody , p = Γ + ˙ q = − mg χ . After this iden tiﬁcation, the hea vy top action principle in Prop osition 23.1 with the Kaluza-Klein Lagrangian returns Euler’s motion equation for the hea vy top (23.1). D. D. Holm Imp erial College London Applications of P oisson Geometry 97 The Hamiltonian H K K asso ciated to L K K b y the Legendre transformation (23.12) is H K K ( Π , Γ , q , p ) = Π · Ω + p · ˙ q − L K K ( Ω , Γ , q , ˙ q ) = 1 2 Π · I − 1 Π − p · Γ + 1 2 | p | 2 = 1 2 Π · I − 1 Π + 1 2 | p − Γ | 2 − 1 2 | Γ | 2 . Recall that Γ is a unit v ector. On the constant lev el set | Γ | 2 = 1, the Kaluza-Klein Hamiltonian H K K is a p ositiv e quadratic function, shifted b y a constant. Likewise, on the constant level set p = − mg χ , the Kaluza-Klein Hamiltonian H K K is a function of only the v ariables ( Π , Γ ) and is equal to the Hamiltonian (23.5) for the heavy top up to an additive constan t. Consequen tly , the Lie-P oisson equations for the Kaluza-Klein Hamiltonian H K K no w repro duce Euler’s motion equation for the hea vy top (23.1). Exercise 23.5 Write the Kaluza-Klein c onstruction on S E (3) = S O (3) s R 3 . 24 Euler-P oincar ´ e (EP) reduction theorem Remark 24.1 (Geo desic motion) As emphasize d by [A r1966], in many inter esting c ases, the Euler– Poinc ar´ e e quations on the dual of a Lie algebr a g ∗ c orr esp ond to ge o desic motion on the c orr esp onding gr oup G . The r elationship b etwe en the e quations on g ∗ and on G is the c ontent of the b asic Euler- Poinc ar´ e the or em discusse d later. Similarly, on the Hamiltonian side, the pr e c e ding p ar agr aphs de- scrib e d the r elation b etwe en the Hamiltonian e quations on T ∗ G and the Lie–Poisson e quations on g ∗ . The issue of ge o desic motion is esp e cial ly simple: if either the L agr angian on g or the Hamiltonian on g ∗ is pur ely quadr atic, then the c orr esp onding motion on the gr oup is ge o desic motion. 24.1 W e w ere already sp eaking prose (EP) Man y of our previous considerations may be recast immediately as Euler-Poincar ´ e equations. • Rigid b odies '  EP S O ( n )  , • Deforming b odies '  EP GL + ( n, R )  , • Hea vy tops '  EP S O (3) × R 3  , • EPDiﬀ 24.2 Euler-P oincar´ e Reduction This lecture applies reduction by symmetry to Hamilton’s principle. F or a G − inv ariant Lagrangian deﬁned on T G , this reduction takes Hamilton’s principle from T G to T G/G ' g . Stationarity of the symmetry-reduced Hamilton’s principle yields the Euler-Poincar ´ e equations on g ∗ . The corresp onding reduced Legendre transformation yields the Lie-P oisson Hamiltonian formulation of these equations. D. D. Holm Imp erial College London Applications of P oisson Geometry 98 Euler-Poinc ar ´ e R e duction starts with a right (resp ectiv ely , left) in v ariant Lagrangian L : T G → R on the tangent bundle of a Lie group G . This means that L ( T g R h ( v )) = L ( v ), resp ec- tiv ely L ( T g L h ( v )) = L ( v ), for all g , h ∈ G and all v ∈ T g G . In shorter notation, right inv ariance of the Lagrangian ma y b e written as L ( g ( t ) , ˙ g ( t )) = L ( g ( t ) h, ˙ g ( t ) h ) , for all h ∈ G . Theorem 24.2 (Euler-Poincar ´ e Reduction) L et G b e a Lie gr oup, L : T G → R a right-invariant L agr angian, and l := L | g : g → R b e its r estriction to g . F or a curve g ( t ) ∈ G , let ξ ( t ) = ˙ g ( t ) · g ( t ) − 1 := T g ( t ) R g ( t ) − 1 ˙ g ( t ) ∈ g . Then the fol lowing four statements ar e e quivalent: (i) g ( t ) satisﬁes the Euler-L agr ange e quations for L agr angian L deﬁne d on G . (ii) The variational principle δ Z b a L ( g ( t ) , ˙ g ( t )) dt = 0 holds, for variations with ﬁxe d endp oints. (iii) The (right invariant) Euler-Poinc ar´ e e quations hold: d dt δ l δ ξ = − ad ∗ ξ δ l δ ξ . (iv) The variational principle δ Z b a l ( ξ ( t )) dt = 0 holds on g , using variations of the form δ ξ = ˙ η − [ ξ , η ] , wher e η ( t ) is an arbitr ary p ath in g which vanishes at the endp oints, i.e., η ( a ) = η ( b ) = 0 . Pro of. Step I. Pr o of that (i) ⇐ ⇒ (ii): This is Hamilton’s principle: the Euler-Lagrange equations follow from stationary action for v ariations δ g which v anish at the endp oin ts. (See Lecture 10.) Step II. Pr o of that (ii) ⇐ ⇒ (iv): Proving equiv alence of the v ariational principles (ii) on T G and (iv) on g for a righ t-inv ariant Lagrangian requires calculation of the v ariations δ ξ of ξ = ˙ g g − 1 induced b y δ g . T o simplify the exp osition, the calculation will b e done ﬁrst for matrix Lie groups, then generalized to arbitrary Lie groups. Step IIA. Pro of that (ii) ⇐ ⇒ (iv) for a matrix Lie group. F or ξ = ˙ g g − 1 , deﬁne g  ( t ) to b e a family of curv es in G such that g 0 ( t ) = g ( t ) and denote δ g := dg  ( t ) d     =0 . D. D. Holm Imp erial College London Applications of P oisson Geometry 99 The v ariation of ξ is computed in terms of δ g as δ ξ = d d      =0 ( ˙ g  g − 1  ) = d 2 g dtd      =0 g − 1 − ˙ g g − 1 ( δ g ) g − 1 . (24.1) Set η := g − 1 δ g . That is, η ( t ) is an arbitrary curve in g whic h v anishes at the endp oin ts. The time deriv ative of η is computed as ˙ η = dη dt = d dt   d d      =0 g   g − 1  = d 2 g dtd      =0 g − 1 − ( δ g ) g − 1 ˙ g g − 1 . (24.2) T aking the diﬀerence of (24.1) and (24.2) implies δ ξ − ˙ η = − ˙ g g − 1 ( δ g ) g − 1 + ( δ g ) g − 1 ˙ g g − 1 = − ξ η + η ξ = − [ ξ , η ] . That is, for matrix Lie algebras, δ ξ = ˙ η − [ ξ , η ] , where [ ξ , η ] is the matrix commutator. Next, we notice that right in v ariance of L allo ws one to change v ariables in the Lagrangian by applying g − 1 ( t ) from the righ t, as L ( g ( t ) , ˙ g ( t )) = L ( e, ˙ g ( t ) g − 1 ( t )) =: l ( ξ ( t )) . Com bining this deﬁnition of the symmetry-reduced Lagrangian l : g → R together with the form ula for v ariations δ ξ just deduced pro v es the equiv alence of (ii) and (iv) for matrix Lie groups. Step I IB. Pro of that (ii) ⇐ ⇒ (iv) for an arbitrary Lie group. The same proof extends to an y Lie group G by using the follo wing lemma. Lemma 24.3 L et g : U ⊂ R 2 → G b e a smo oth map and denote its p artial derivatives by ξ ( t, ε ) := T g ( t,ε ) R g ( t,ε ) − 1 ∂ g ( t, ε ) ∂ t , η ( t, ε ) := T g ( t,ε ) R g ( t,ε ) − 1 ∂ g ( t, ε ) ∂ ε . (24.3) Then ∂ ξ ∂ ε − ∂ η ∂ t = − [ ξ , η ] , (24.4) wher e [ ξ , η ] is the Lie algebr a br acket on g . Conversely, if U ⊂ R 2 is simply c onne cte d and ξ , η : U → g ar e smo oth functions satisfying (24.4) , then ther e exists a smo oth function g : U → G such that (24.3) holds. Pro of. W rite ξ = ˙ g g − 1 and η = g 0 g − 1 in natural notation and express the partial deriv atives ˙ g = ∂ g /∂ t and g 0 = ∂ g /∂  using the righ t translations as ˙ g = ξ ◦ g and g 0 = η ◦ g . By the c hain rule, these deﬁnitions hav e mixed partial deriv atives ˙ g 0 = ξ 0 = ∇ ξ · η and ˙ g 0 = ˙ η = ∇ η · ξ . The diﬀerence of the mixed partial deriv atives implies the desired formula (24.4), ξ 0 − ˙ η = ∇ ξ · η − ∇ η · ξ = − [ ξ , η ] = − ad ξ η . (Note the min us sign in the last tw o terms.) D. D. Holm Imp erial College London Applications of P oisson Geometry 100 Step III. Pr o of of e quivalenc e (iii) ⇐ ⇒ (iv) Let us show that the reduced v ariational principle pro duces the Euler-Poincar ´ e equations. W e write the functional deriv ative of the reduced action S red = R b a l ( ξ ) dt with Lagrangian l ( ξ ) in terms of the natural pairing h· , ·i b et ween g ∗ and g as δ Z b a l ( ξ ( t )) dt = Z b a  δ l δ ξ , δ ξ  dt = Z b a  δ l δ ξ , ˙ η − ad ξ η  dt = Z b a  δ l δ ξ , ˙ η  dt − Z b a  δ l δ ξ , ad ξ η  dt = − Z b a  d dt δ l δ ξ + ad ∗ ξ δ l δ ξ , η  dt . The last equality follows from in tegration b y parts and v anishing of the v ariation η ( t ) at the endpoints. Th us, stationarity δ R b a l ( ξ ( t )) dt = 0 for any η ( t ) that v anishes at the endp oin ts is equiv alent to d dt δ l δ ξ = − ad ∗ ξ δ l δ ξ , whic h are the Euler-Poincar ´ e equations. Remark 24.4 (Left-inv arian t Euler-P oincar´ e equations) The same the or em holds for left in- variant L agr angians on T G , exc ept for a sign in the Euler-Poinc ar ´ e e quations, d dt δ l δ ξ = + ad ∗ ξ δ l δ ξ , which arises b e c ause left-invariant variations satisfy δ ξ = ˙ η + [ ξ , η ] (with the opp osite sign). Exercise 24.5 Write out the c orr esp onding pr o of of the Euler-Poinc ar´ e r e duction the or em for left- invariant L agr angians deﬁne d on the tangent sp ac e T G of a gr oup G . Reconstruction. The pro cedure for reconstructing the solution v ( t ) ∈ T g ( t ) G of the Euler-Lagrange equations with initial conditions g (0) = g 0 and ˙ g (0) = v 0 starting from the solution of the Euler- P oincar´ e equations is as follows. First, solve the initial v alue problem for the right-in v ariant Euler- P oincar´ e equations: d dt δ l δ ξ = − ad ∗ ξ δ l δ ξ with ξ (0) = ξ 0 := v 0 g − 1 0 . Then from the solution for ξ ( t ) reconstruct the curv e g ( t ) on the group by solving the “linear diﬀerential equation with time-dep enden t co eﬃcien ts” ˙ g ( t ) = ξ ( t ) g ( t ) with g (0) = g 0 . The Euler-Poincar ´ e reduction theorem guarantees then that v ( t ) = ˙ g ( t ) = ξ ( t ) · g ( t ) is a solution of the Euler-Lagrange equations with initial condition v 0 = ξ 0 g 0 . Remark 24.6 A similar statement holds, with obvious changes for left-invariant L agr angian systems on T G . D. D. Holm Imp erial College London Applications of P oisson Geometry 101 24.3 Reduced Legendre transformation As in the equiv alence relation b et ween the Lagrangian and Hamiltonian form ulations discussed earlier, the relationship betw een symmetry-reduced Euler-P oincar´ e and Lie-P oisson formulations is determined b y the Legendre transformation. Deﬁnition 24.7 The L e gendr e tr ansformation F l : g → g ∗ is deﬁne d by F l ( ξ ) = δ l δ ξ = µ . Lie-P oisson Hamiltonian form ulation. Let h ( µ ) := h µ, ξ i − l ( ξ ). Assuming that F l is a diﬀeo- morphism yields δ h δ µ = ξ +  µ , δ ξ δ µ  −  δ l δ ξ , δ ξ δ µ  = ξ . So the Euler-P oincar´ e equations for l are equiv alen t to the Lie-Poisson equations for h : d dt  δ l δ ξ  = − ad ∗ ξ δ l δ ξ ⇐ ⇒ ˙ µ = − ad ∗ δ h/δµ µ. The Lie-P oisson equations may be written in the Poisson brac k et form ˙ f = { f , h } , (24.5) where f : g ∗ → R is an arbitrary smo oth function and the brac ket is the (right) Lie-Poisson brack et giv en by { f , h } ( µ ) =  µ,  δ f δ µ , δ h δ µ  = −  µ, ad δ h/δµ δ f δ µ  = −  ad ∗ δ h/δµ µ, δ f δ µ  . (24.6) In the imp ortan t case when ` is quadratic, the Lagrangian L is the quadratic form asso ciated to a right in v ariant Riemannian metric on G . In this case, the Euler–Lagrange equations for L on G describ e geo desic motion relativ e to this metric and these geodesics are then equiv alen tly describ ed b y either the Euler-Poincar ´ e, or the Lie-Poisson equations. Exercise 24.8 Exer cise 22.6 r e quir es an extension of the pur e EP r e duction the or em for a L agr angian L : ( T G × T Q ) → R . F ol lowing the pr o of of the EP r e duction the or em, make this extension. Exercise 24.9 Compute the pur e EP e quations for ge o desic motion on S E (3) . These e quations turn out to b e applic able to the motion of an el lipsoidal b o dy thr ough a ﬂuid. 25 EPDiﬀ: the Euler-P oincar ´ e equation on the diﬀeomorphisms 25.1 The n − dimensional EPDiﬀ equation and its prop erties Eulerian geo desic motion of a ﬂuid in n − dimensions is generated as an EP equation via Hamilton’s principle, when the Lagrangian is given by the kinetic energy . The kinetic energy deﬁnes a norm k u k 2 for the Eulerian ﬂuid v elo cit y , tak en as u ( x , t ) : R n × R 1 → R n . The c hoice of the kinetic energy as a p ositiv e functional of ﬂuid velocity u is a mo deling step that dep ends up on the physics of the problem b eing studied. W e shall choose the Lagrangian, k u k 2 = Z u · Q op u d n x = Z u · m d n x , (25.1) D. D. Holm Imp erial College London Applications of P oisson Geometry 102 so that the p ositive-deﬁnite, symmetric, operator Q op deﬁnes the norm k u k , for appropriate (homo- geneous, say , or p erio dic) b oundary conditions. The EPDiﬀ equation is the Euler-P oincar´ e equation for this Eulerian geo desic motion of a ﬂuid. Namely , d dt δ ` δ u + ad ∗ u δ ` δ u = 0 , with ` [ u ] = 1 2 k u k 2 . (25.2) Here ad ∗ is the dual of the v ector-ﬁeld ad-operation (the commutator) under the natural L 2 pairing h· , ·i induced b y the v ariational deriv ative δ ` [ u ] = h δ `/δ u , δ u i . This pairing provides the deﬁnition of ad ∗ , h ad ∗ u m , v i = − h m , ad u v i , (25.3) where u and v are vector ﬁelds, ad u v = [ u , v ] is the commutator, i.e., the Lie br acket given in comp onen ts b y (summing on rep eated indices) [ u , v ] i = u j ∂ v i ∂ x j − v j ∂ u i ∂ x j , or [ u , v ] = u · ∇ v − v · ∇ u . (25.4) The notation ad u v := [ u , v ] formally denotes the adjoint action of the right Lie algebra of Diﬀ ( D ) on itself, and m = δ `/δ u is the ﬂuid momentum, a one-form density whose co-v ector comp onen ts are also denoted as m . If u = u j ∂ /∂ x j , m = m i dx i ⊗ dV , then the preceding form ula for ad ∗ u ( m ⊗ dV ) has the c o or dinate expr ession in R n ,  ad ∗ u m  i dx i ⊗ dV =  ∂ ∂ x j ( u j m i ) + m j ∂ u j ∂ x i  dx i ⊗ dV . (25.5) In this notation, the abstract EPDiﬀ equation (25.2) may b e written explicitly in Euclidean coor- dinates as a partial diﬀeren tial equation for a co-vector function m ( x , t ) : R n × R 1 → R n . Namely , ∂ ∂ t m + u · ∇ m | {z } Con vection + ∇ u T · m | {z } Stretc hing + m (div u ) | {z } Expansion = 0 , with m = δ ` δ u = Q op u . (25.6) T o explain the terms in underbraces, we rewrite EPDiﬀ as preserv ation of the one-form density of momen tum along the characteristic curv es of the v elo cit y . Namely , d dt  m · d x ⊗ dV  = 0 along d x dt = u = G ∗ m . (25.7) This form of the EPDiﬀ equation also emphasizes its nonlo calit y , since the velocity is obtained from the momen tum density by conv olution against the Green’s function G of the op erator Q op . Th us, u = G ∗ m with Q op G = δ ( x ), the Dirac measure. W e ma y c hec k that this “c haracteristic form” of EPDiﬀ reco vers its Eulerian form b y computing directly , d dt  m · d x ⊗ dV  = d m dt · d x ⊗ dV + m · d d x dt ⊗ dV + m · d x ⊗  d dt dV  along d x dt = u = G ∗ m =  ∂ ∂ t m + u · ∇ m + ∇ u T · m + m (div u )  · d x ⊗ dV = 0 . Exercise 25.1 Show that EPDiﬀ may b e written as  ∂ ∂ t + L u  m · d x ⊗ dV  = 0 , (25.8) wher e L u is the Lie derivative with r esp e ct to the ve ctor ﬁeld with c omp onents u = G ∗ m . Hint: How do es this Lie-derivative form of EPDiﬀ in (25.8) diﬀer fr om its char acteristic form (25.7)? D. D. Holm Imp erial College London Applications of P oisson Geometry 103 EPDiﬀ ma y also b e written equiv alen tly in terms of the operators div, grad and curl in 2D and 3D as, ∂ ∂ t m − u × curl m + ∇ ( u · m ) + m (div u ) = 0 . (25.9) Th us, for example, its numerical solution w ould require an algorithm which has the capabilit y to deal with the distinctions and relationships among the op erators div, grad and curl. 25.2 Deriv ation of the n − dimensional EPDiﬀ equation as geo desic ﬂo w Let’s derive the EPDiﬀ equation (25.6) by following the proof of the EP reduction theorem leading to the Euler-P oincar´ e equations for right inv ariance in the form (25.2). F ollo wing this calculation for the presen t case yields δ Z b a l ( u ) dt = Z b a  δ l δ u , δ u  dt = Z b a  δ l δ u , ˙ v − ad u v  dt = Z b a  δ l δ u , ˙ v  dt − Z b a  δ l δ u , ad u v  dt = − Z b a  d dt δ l δ u + ad ∗ u δ l δ u , v  dt , where h· , ·i is the pairing b et ween elements of the Lie algebra and its dual. In our case, this is the L 2 pairing, e.g.,  δ l δ u , δ u  = Z δ l δ u i δ u i d n x This pairing allo ws us to compute the co ordinate form of the EPDiﬀ equation explicitly , as Z b a  δ l δ u , δ u  dt = Z b a dt Z δ l δ u i  ∂ v i ∂ t + u j ∂ v i ∂ x j − v j ∂ u i ∂ x j  d n x = − Z b a dt Z  ∂ ∂ t δ l δ u i + ∂ ∂ x j  δ l δ u i u j  + δ l δ u j ∂ u j ∂ x i  v i d n x Substituting m = δ l /δ u no w recov ers the co ordinate forms for the coadjoin t action of vector ﬁelds in (25.5) and the EPDiﬀ equation itself in (25.6). When ` [ u ] = 1 2 k u k 2 , EPDiﬀ describ es geo desic motion on the diﬀeomorphisms with resp ect to the norm k u k . Lemma 25.2 In Step IIB of the pr o of of the Euler-Poinc ar´ e r e duction the or em that (ii) ⇐ ⇒ (iv) for an arbitr ary Lie gr oup, a c ertain formula for the variations for time-dep endent ve ctor ﬁelds was employe d. That formula was employe d again in the c alculation ab ove as, δ u = ˙ v − ad u v . (25.10) This formula may b e r e derive d as fol lows in the pr esent c ontext. We write u = ˙ g g − 1 and v = g 0 g − 1 in natur al notation and expr ess the p artial derivatives ˙ g = ∂ g /∂ t and g 0 = ∂ g /∂  using the right tr anslations as ˙ g = u ◦ g and g 0 = v ◦ g . T o c ompute the mixe d p artials, c onsider the chain rule for say u ( g ( t,  ) x 0 ) and set x ( t,  ) = g ( t,  ) · x 0 . Then, u 0 = ∂ u ∂ x · ∂ x ∂  = ∂ u ∂ x · g 0 ( t,  ) x 0 = ∂ u ∂ x · g 0 g − 1 x = ∂ u ∂ x · v ( x ) . D. D. Holm Imp erial College London Applications of P oisson Geometry 104 The chain rule for ˙ v gives a similar formula with u and v exchange d. Thus, the chain rule gives two expr essions for the mixe d p artial derivative ˙ g 0 as ˙ g 0 = u 0 = ∇ u · v and ˙ g 0 = ˙ v = ∇ v · u . The diﬀer enc e of the mixe d p artial derivatives then implies the desir e d formula (25.10), sinc e u 0 − ˙ v = ∇ u · v − ∇ v · u = − [ u , v ] = − ad u v . 26 EPDiﬀ: the Euler-P oincar ´ e equation on the diﬀeomorphisms In this lecture, w e shall discuss the solutions of EPDiﬀ for pressureless compressible geo desic motion in one spatial dimension. This is the EPDiﬀ e quation in 1D, 9 ∂ t m + ad ∗ u m = 0 , or, equiv alen tly , (26.1) ∂ t m + um x + 2 u x m = 0 , with m = Q op u . (26.2) • The EPDiﬀ equation describ es geo desic motion on the diﬀeomorphism group with resp ect to a family of metrics for the ﬂuid v elo cit y u ( t, x ), with notation, m = δ ` δ u = Q op u for a kinetic-energy Lagrangian (26.3) ` ( u ) = 1 2 Z u Q op u dx = 1 2 k u k 2 . (26.4) • In one dimension, Q op in equation (26.3) is a p ositive, symmetric operator that deﬁnes the kinetic energy metric for the v elo cit y . • The EPDiﬀ equation (26.2) is written in terms of the v ariable m = δ `/δ u . It is appropriate to call this v ariational deriv ative m , because it is the momen tum density associated with the ﬂuid v elo cit y u . • Ph ysically , the ﬁrst nonlinear term in the EPDiﬀ equation (26.2) is ﬂuid transp ort. • The co eﬃcient 2 arises in the second nonlinear term, b ecause, in one dimension, tw o of the summands in ad ∗ u m = um x + 2 u x m are the same , cf. equation (25.5). • The momentum is expressed in terms of the v elo cit y b y m = δ `/δ u = Q op u . Equiv alently , for solutions that v anish at spatial inﬁnit y , one ma y think of the v elo city as b eing obtained from the con volution, u ( x ) = G ∗ m ( x ) = Z G ( x − y ) m ( y ) dy , (26.5) where G is the Green’s function for the op erator Q op on the real line. • The op erator Q op and its Green’s function G are chosen to be even under reﬂection, G ( − x ) = G ( x ), so that u and m ha ve the same parit y . Moreov er, the EPDiﬀ equation (26.2) conserv es the total momen tum M = R m ( y ) dy , for an y even Green’s function. 9 A one-form density in 1D takes the form m ( dx ) 2 and the EP equation is giv en b y d dt ` m ( dx ) 2 ´ = dm dt ( dx ) 2 + 2 m ( du )( dx ) = 0 with d dt dx = du = u x dx and u = G ∗ m , where G ∗ m denotes con volution with a function G on the real line. D. D. Holm Imp erial College London Applications of P oisson Geometry 105 Exercise 26.1 Show that e quation (26.2) c onserves M = R m ( y ) dy for any even Gr e en ’s func- tion G ( − x ) = G ( x ) , for either p erio dic, or homo gene ous b oundary c onditions. • The tra veling w a ve solutions of 1D EPDiﬀ when the Green’s function G is chosen to be even under reﬂection are the “pulsons,” u ( x, t ) = c G ( x − ct ) . Exercise 26.2 Pr ove this statement, that the tr aveling wave solutions of 1D EPDiﬀ ar e pulsons when the Gr e en ’s function is even. What r ole is playe d in the solution by the Gr e en ’s function b eing even? Hint: Evaluate the derivative of an even function at x = 0 . • See F ringer and Holm F rHo2001 and references therein for further discussions and numerical sim ulations of the pulson solutions of the 1D EPDiﬀ equation. 26.1 Pulsons The EPDiﬀ equation (26.2) on the real line has the remark able prop ert y that its solutions c ol le c- tivize 10 in to the ﬁnite dimensional solutions of the “ N − pulson” form that was discov ered for a special form of G in Camassa and Holm CaHo1993, then was extended for any even G in F ringer and Holm F rHo2001, u ( x, t ) = N X i =1 p i ( t ) G ( x − q i ( t )) . (26.6) Since G ( x ) is the Green’s function for the op erator Q op , the corresp onding solution for the momentum m = Q op u is giv en by a sum of delta functions, m ( x, t ) = N X i =1 p i ( t ) δ ( x − q i ( t )) . (26.7) Th us, the time-dep enden t “collective co ordinates” q i ( t ) and p i ( t ) are the p ositions and velocities of the N pulses in this solution. These parameters satisfy the ﬁnite dimensional geo desic motion equations obtained as canonical Hamiltonian equations ˙ q i = ∂ H N ∂ p i = N X j =1 p j G ( q i − q j ) , (26.8) ˙ p i = − ∂ H N ∂ q i = − p i N X j =1 p j G 0 ( q i − q j ) , (26.9) in whic h the Hamiltonian is given b y the quadratic form, H N = 1 2 N X i,j =1 p i p j G ( q i − q j ) . (26.10) 10 See [GuSt1984] for discussions of the concept of collective v ariables for Hamiltonian theories. W e will discuss the collectivization for the EPDiﬀ equation later from the viewp oin t of momentum maps. D. D. Holm Imp erial College London Applications of P oisson Geometry 106 Remark 26.3 In a c ertain sense, e quations (26.8-26.9) c omprise the analo g for the p e akon momen- tum r elation (26.7) of the “symmetric gener alize d rigid b o dy e quations” in (20.1). Th us, the canonical equations for the Hamiltonian H N describ e the nonlinear collective in teractions of the N − pulson solutions of the EPDiﬀ equation (26.2) as ﬁnite-dimensional geodesic motion of a particle on an N − dimensional surface whose co-metric is G ij ( q ) = G ( q i − q j ) . (26.11) F ringer and Holm F rHo2001 sho wed numerically that the N − pulson solutions describ e the emergent patterns in the solution of the initial v alue problem for EPDiﬀ equation (26.2) with spatially conﬁned initial conditions. Exercise 26.4 Equations (26.8-26.9) describ e ge o desic motion. 1. Write the L agr angian and Euler-L agr ange e quations for this motion. 2. Solve e quations (26.8-26.9) for N = 2 when lim | x |→∞ G ( x ) = 0 . (a) Why should the solution b e describ e d as exchange of momentum in elastic c ol lisions? (b) Consider b oth he ad-on and overtaking c ol lisions. (c) Consider the antisymmetric c ase, when the total momentum vanishes. In tegrabilit y Calogero and F rancoise [Ca1995], [CaF r1996] found that for any ﬁnite num b er N the Hamiltonian equations for H N in (26.10) are completely in tegrable in the Liouville sense 11 for G ≡ G 1 ( x ) = λ + µ cos( ν x ) + µ 1 sin( ν | x | ) and G ≡ G 2 ( x ) = α + β | x | + γ x 2 , with λ , µ , µ 1 , ν , and α , β , γ b eing arbitrary constants, suc h that λ and µ are real and µ 1 and ν b oth real or b oth imaginary . 12 P articular cases of G 1 and G 2 are the p eak ons G 1 ( x ) = e −| x | /α of [CaHo1993] and the compactons G 2 ( x ) = max(1 − | x | , 0) of the Hun ter-Saxton equation, [HuZh1994]. The latter is the EPDiﬀ equation (26.2), with ` ( u ) = 1 2 R u 2 x dx and th us m = − u xx . Lie-P oisson Hamiltonian form of EPDiﬀ In terms of m , the conserv ed energy Hamiltonian for the EPDiﬀ equation (26.2) is obtained b y Legendre transforming the kinetic energy Lagrangian, as h = D δ ` δ u , u E − ` ( u ) . Th us, the Hamiltonian dep ends on m , as h ( m ) = 1 2 Z m ( x ) G ( x − y ) m ( y ) dxdy , whic h also reveals the geodesic nature of the EPDiﬀ equation (26.2) and the role of G ( x ) in the kinetic energy metric on the Hamiltonian side. The corresp onding Lie-Poisson br acket for EPDiﬀ as a Hamiltonian evolution equation is given b y , ∂ t m =  m, h  = − ( ∂ m + m∂ ) δ h δ m and δ h δ m = u , 11 A Hamiltonian system is integrable in the Liouville sense, if the n umber of indep enden t constan ts of motion in in volution is the same as the num b er of its degrees of freedom. 12 This choice of the constants keeps H N real in (26.10). D. D. Holm Imp erial College London Applications of P oisson Geometry 107 whic h reco v ers the starting equation and indicates some of its connections with ﬂuid equations on the Hamiltonian side. F or an y t wo smo oth functionals f , h of m in the space for which the solutions of EPDiﬀ exist, this Lie-P oisson brack et ma y b e expressed as,  f , h  = − Z δ f δ m ( ∂ m + m∂ ) δ h δ m dx = − Z m  δ f δ m , δ h δ m  dx where [ · , · ] denotes the Lie algebra brack et of v ector ﬁelds. That is,  δ f δ m , δ h δ m  = δ f δ m ∂ δ h δ m − δ h δ m ∂ δ f δ m . Exercise 26.5 What is the Casimir for this Lie Poisson br acket? What do es it me an fr om the view- p oint of c o adjoint orbits? 26.2 P eak ons The case G ( x ) = e −| x | /α with a constan t lengthscale α is the Green’s function for whic h the op erator in the kinetic energy Lagrangian (26.3) is Q op = 1 − α 2 ∂ 2 x . F or this (Helmholtz) operator Q op , the Lagrangian and corresp onding kinetic energy norm are giv en b y , ` [ u ] = 1 2 k u k 2 = 1 2 Z u Q op u dx = 1 2 Z u 2 + α 2 u 2 x dx , for lim | x |→∞ u = 0 . This Lagrangian is the H 1 norm of the velocity in one dimension. In this case, the EPDiﬀ equation (26.2) is also the zero-disp ersion limit of the completely integrable CH equation for unidirectional shallo w water w av es ﬁrst derived in Camassa and Holm CaHo1993, m t + um x + 2 mu x = − c 0 u x + γ u xxx , m = u − α 2 u xx . (26.12) This equation describes shallow w ater dynamics as completely in tegrable soliton motion at quadratic order in the asymptotic expansion for unidirectional shallow water wa ves on a free surface under gra vity . See Dullin, Gott wald and Holm DGH[2001,2003,2004] for more details and explanations of this asymptotic expansion for unidirectional shallo w water w a ves to quadratic order. Because of the relation m = u − α 2 u xx , equation (26.12) is nonlo cal. In other words, it is an in tegral-partial diﬀerential equation. In fact, after writing equation (26.12) in the equiv alent form, (1 − α 2 ∂ 2 )( u t + uu x ) = − ∂  u 2 + α 2 2 u 2 x  − c 0 u x + γ u xxx , (26.13) one sees the in terplay betw een lo cal and nonlocal linear disp ersion in its phase v elo cit y relation, ω k = c 0 − γ k 2 1 + α 2 k 2 , (26.14) for wa ves with frequency ω and wa ve num b er k linearized around u = 0. F or γ /c 0 < 0, short w a ves and long wa ves tra v el in the same direction. Long w av es trav el faster than short ones (as required in shallo w water) provided γ /c 0 > − α 2 . Then the phase v elo cit y lies in the interv al ω /k ∈ ( − γ /α 2 , c 0 ]. The famous Kortew eg-de V ries (KdV) soliton equation, u t + 3 uu x = − c 0 u x + γ u xxx , (26.15) D. D. Holm Imp erial College London Applications of P oisson Geometry 108 emerges at line ar order in the asymptotic expansion for shallo w water w av es, in which one takes α 2 → 0 in (26.13) and (26.14). In KdV, the parameters c 0 and γ are seen as deformations of the R iemann e quation , u t + 3 uu x = 0 . The parameters c 0 and γ represent linear wa ve disp ersion, which mo diﬁes and even tually balances the tendency for nonlinear wa ves to steep en and break. The parameter α , whic h introduces nonlo cality , also regularizes this nonlinear tendency , even in the absence of c 0 and γ . 27 Diﬀeons – singular momen tum solutions of the EPDiﬀ equation for geo desic motion in higher dimensions As an example of the EP theory in higher dimensions, we shall generalize the one-dimensional pulson solutions of the previous section to n − dimensions. The corresp onding singular momen tum solutions of the EPDiﬀ equation in higher dimensions are called “diﬀeons.” 27.1 n − dimensional EPDiﬀ equation Eulerian geo desic motion of a ﬂuid in n − dimensions is generated as an EP equation via Hamilton’s principle, when the Lagrangian is given by the kinetic energy . The kinetic energy deﬁnes a norm k u k 2 for the Eulerian ﬂuid velocity , u ( x , t ) : R n × R 1 → R n . As mentioned earlier, the c hoice of the kinetic energy as a positive functional of ﬂuid v elo city u is a modeling step that dep ends up on the ph ysics of the problem being studied. F ollowing our earlier pro cedure, as in equations (25.1) and (25.2), we shall c ho ose the Lagrangian, k u k 2 = Z u · Q op u d n x = Z u · m d n x , (27.1) so that the p ositiv e-deﬁnite, symmetric, op erator Q op deﬁnes the norm k u k , for appropriate b oundary conditions and the EPDiﬀ equation for Eulerian geo desic motion of a ﬂuid emerges, d dt δ ` δ u + ad ∗ u δ ` δ u = 0 , with ` [ u ] = 1 2 k u k 2 . (27.2) Legendre transforming to the Hamiltonian side The corresp onding Legendre transform yields the follo wing inv ertible relations b et ween momen tum and velocity , m = Q op u and u = G ∗ m , (27.3) where G is the Green’s function for the op erator Q op , assuming appropriate boundary conditions (on u ) that allo w inv ers ion of the op erator Q op to determine u from m . The corresp onding Hamiltonian is, h [ m ] = h m , u i − 1 2 k u k 2 = 1 2 Z m · G ∗ m d n x ≡ 1 2 k m k 2 , (27.4) whic h also deﬁnes a norm k m k via a conv olution k ernel G that is symmetric and p ositive, when the Lagrangian ` [ u ] is a norm. As exp ected, the norm k m k given by the Hamiltonian h [ m ] sp eciﬁes the v elo cit y u in terms of its Legendre-dual momentum m b y the v ariational op eration, u = δ h δ m = G ∗ m ≡ Z G ( x − y ) m ( y ) d n y . (27.5) D. D. Holm Imp erial College London Applications of P oisson Geometry 109 W e shall c ho ose the kernel G ( x − y ) to b e translation-in v ariant (so No ether’s theorem implies that total momentum M = R m d n x is conserved) and symmetric under spatial reﬂections (so that u and m hav e the same parit y). After the Legendre transformation (27.4), the EPDiﬀ equation (27.2) app ears in its equiv alen t Lie-P oisson Hamiltonian form , ∂ ∂ t m = { m , h } = − ad ∗ δ h/δ m m . (27.6) Here the operation {· , · } denotes the Lie-P oisson brac ket dual to the (righ t) action of v ector ﬁelds amongst themselv es by v ector-ﬁeld comm utation { f , h } = −  m ,  δ f δ m , δ h δ m  F or more details and additional background concerning the relation of classical EP theory to Lie- P oisson Hamiltonian equations, see [MaRa1994, HoMaRa1998a]. In a momen t we will also consider the momen tum maps for EPDiﬀ. 27.2 Diﬀeons: n − dimensional Analogs of Pulsons for the EPDiﬀ equation The momentum for the one-dimensional pulson solutions ( ?? ) on the real line is supp orted at p oints via the Dirac delta measures in its solution ansatz, m ( x, t ) = N X i =1 p i ( t ) δ  x − q i ( t )  , m ∈ R 1 . (27.7) W e shall develop n − dimensional analogs of these one-dimensional pulson solutions for the Euler- P oincar´ e equation (25.9) by generalizing this solution ansatz to allo w measure-v alued n − dimensional v ector solutions m ∈ R n for whic h the Euler-Poincar ´ e momen tum is supp orted on co-dimension − k subsp ac es R n − k with integer k ∈ [1 , n ]. F or example, one may consider the tw o-dimensional v ector momen tum m ∈ R 2 in the plane that is supp orted on one-dimensional curves (momen tum fron ts). Lik ewise, in three dimensions, one could consider tw o-dimensional momen tum s urfaces (sheets), one- dimensional momentum ﬁlamen ts, etc. The corresp onding vector momentum ansatz that w e shall use is the follo wing, cf. the pulson solutions (27.7), m ( x , t ) = N X i =1 Z P i ( s, t ) δ  x − Q i ( s, t )  ds , m ∈ R n . (27.8) Here, P i , Q i ∈ R n for i = 1 , 2 , . . . , N . F or example, when n − k = 1, so that s ∈ R 1 is one- dimensional, the delta function in solution (27.8) supports an evolving family of vector-v alued curv es, called momentum ﬁlaments . (F or simplicit y of notation, we suppress the implied subscript i in the arclength s for eac h P i and Q i .) The Legendre-dual relations (27.3) imply that the velocity corresp onding to the momen tum ﬁlamen t ansatz (27.8) is, u ( x , t ) = G ∗ m = N X j =1 Z P j ( s 0 , t ) G  x − Q j ( s 0 , t )  ds 0 . (27.9) Just as for the 1D case of the pulsons, we shall show that substitution of the n − D solution ansatz (27.8) and (27.9) in to the EPDiﬀ equation (25.6) pro duces canonical geo desic Hamiltonian equations for the n − dimensional v ector parameters Q i ( s, t ) and P i ( s, t ), i = 1 , 2 , . . . , N . D. D. Holm Imp erial College London Applications of P oisson Geometry 110 27.2.1 Canonical Hamiltonian dynamics of diﬀeon momentum ﬁlaments in R n F or deﬁniteness in what follo ws, w e shall consider the example of momen tum ﬁlaments m ∈ R n supp orted on one-dimensional space curves in R n , so s ∈ R 1 is the arclength parameter of one of these curv es. This solution ansatz is reminiscent of the Biot-Sav art Law for v ortex ﬁlamen ts, although the ﬂo w is not incompressible. The dynamics of momentum surfaces, for s ∈ R k with k < n , follo w a similar analysis. Substituting the momen tum ﬁlamen t ansatz (27.8) for s ∈ R 1 and its corresponding velocity (27.9) in to the Euler-Poincar ´ e equation (25.6), then in tegrating against a smo oth test function φ ( x ) implies the follo wing canonical equations (denoting explicit summation on i, j ∈ 1 , 2 , . . . N ), ∂ ∂ t Q i ( s, t ) = N X j =1 Z P j ( s 0 , t ) G ( Q i ( s, t ) − Q j ( s 0 , t ))  ds 0 = δ H N δ P i , (27.10) ∂ ∂ t P i ( s, t ) = − N X j =1 Z  P i ( s, t ) · P j ( s 0 , t )  ∂ ∂ Q i ( s, t ) G  Q i ( s, t ) − Q j ( s 0 , t )  ds 0 = − δ H N δ Q i , (sum on j , no sum on i ) . (27.11) The dot pro duct P i · P j denotes the inner, or scalar, pro duct of the tw o v ectors P i and P j in R n . Thus, the solution ansatz (27.8) yields a closed set of inte gr o-p artial-diﬀer ential e quations (IPDEs) giv en by (27.10) and (27.11) for the v ector parameters Q i ( s, t ) and P i ( s, t ) with i = 1 , 2 . . . N . These equations are generated canonically b y the following Hamiltonian function H N : ( R n × R n ) ⊗ N → R , H N = 1 2 Z Z N X i , j =1  P i ( s, t ) · P j ( s 0 , t )  G  Q i ( s, t ) − Q j ( s 0 , t )  ds ds 0 . (27.12) This Hamiltonian arises by substituting the momentum ansatz (27.8) into the Hamiltonian (27.4) obtained from the Legendre transformation of the Lagrangian corresp onding to the kinetic energy norm of the ﬂuid velocity . Th us, the evolutionary IPDE system (27.10) and (27.11) represents canonically Hamiltonian geo desic motion on the space of curv es in R n with resp ect to the co-metric giv en on these curv es in (27.12). The Hamiltonian H N = 1 2 k P k 2 in (27.12) deﬁnes the norm k P k in terms of this co-metric that com bines con volution using the Green’s function G and sum ov er ﬁlaments with the scalar pro duct of momen tum v ectors in R n . Remark 27.1 Note the L agr angian pr op erty of the s c o or dinate, sinc e ∂ ∂ t Q i ( s, t ) = u ( Q i ( s, t ) , t ) . 28 Singular solution momen tum map J Sing for diﬀeons The diﬀeon momentum ﬁlament ansatz (27.8) reduces, and c ol le ctivizes the solution of the geo desic EP PDE (25.6) in n + 1 dimensions in to the s ystem (27.10) and (27.11) of 2 N canonical evolutionary IPDEs. One can summarize the mec hanism by whic h this process o ccurs, by sa ying that the map that implemen ts the canonical ( Q , P ) v ariables in terms of singular solutions is a (cotangent bundle) momen tum map. Such momentum maps are P oisson maps; so the canonical Hamiltonian nature of the dynamical equations for ( Q , P ) ﬁts into a general theory which also pro vides a framework for suggesting other a ven ues of in vestigation. D. D. Holm Imp erial College London Applications of P oisson Geometry 111 Theorem 28.1 The momentum ansatz (27.8) for me asur e-value d solutions of the EPDiﬀ e quation (25.6) , deﬁnes an e quivariant momentum map J Sing : T ∗ Em b( S, R n ) → X ( R n ) ∗ that is c al le d the singular solution momentum map in [HoMa2004]. W e shall explain the notation used in the theorem’s statemen t in the course of its pro of. Righ t a wa y , how ever, w e note that the sense of “deﬁnes” is that the momen tum solution ansatz (27.8) expressing m (a v ector function of spatial position x ) in terms of Q , P (which are functions of s ) can b e regarded as a map from the space of ( Q ( s ) , P ( s )) to the space of m ’s. This will turn out to b e the Lagrange-to-Euler map for the ﬂuid description of the singular solutions. F ollowing [HoMa2004], we shall give t wo pro ofs of this result from tw o rather diﬀerent viewp oin ts. The ﬁrst pro of b elo w uses the form ula for a momen tum map for a cotangent lifted action, while the second pro of fo cuses on a Poisson brac ket computation. Eac h pro of also explains the context in which one has a momen tum map. (See [MaRa1994] for general background on momentum maps.) First Pro of. F or simplicit y and without loss of generalit y , let us take N = 1 and so suppress the index a . That is, w e shall tak e the case of an isolated singular solution. As the pro of will show, this is not a real restriction. T o set the notation, ﬁx a k -dimensional manifold S with a given v olume element and whose p oin ts are denoted s ∈ S . Let Emb( S, R n ) denote the set of smo oth embeddings Q : S → R n . (If the EPDiﬀ equations are tak en on a manifold M , replace R n with M .) Under appropriate technical conditions, whic h w e shall just treat formally here, Emb( S, R n ) is a smo oth manifold. (See, for example, [EbMa1970] and [MaHu1983] for a discussion and references.) The tangent space T Q Em b( S, R n ) to Emb( S, R n ) at the p oint Q ∈ Em b( S , R n ) is given b y the space of material velo city ﬁelds , namely the linear space of maps V : S → R n that are v ector ﬁelds o ver the map Q . The dual space to this space will b e identiﬁed with the space of one-form densities ov er Q , whic h w e shall regard as maps P : S → ( R n ) ∗ . In summary , the cotangent bundle T ∗ Em b( S, R n ) is iden tiﬁed with the space of pairs of maps ( Q , P ). These give us the domain space for the singular solution momen tum map. No w w e consider the action of the symmetry group. Consider the group G = Diﬀ of diﬀeomorphisms of the space S in whic h the EPDiﬀ equations are op erating, concretely in our case R n . Let it act on S b y comp osition on the left . Namely for η ∈ Diﬀ ( R n ), w e let η · Q = η ◦ Q . (28.1) No w lift this action to the cotangen t bundle T ∗ Em b( S, R n ) in the standard wa y (see, for instance, MaRa1994 for this construction). This lifted action is a symplectic (and hence P oisson) action and has an equiv ariant momentum map. We claim that this momentum map is pr e cisely given by the ansatz (27.8) . T o see this, one only needs to recall and then apply the general formula for the momentum map asso ciated with an action of a general Lie group G on a conﬁguration manifold Q and cotangent lifted to T ∗ Q . First let us recall the general formula. Namely , the momentum map is the map J : T ∗ Q → g ∗ ( g ∗ denotes the dual of the Lie algebra g of G ) deﬁned b y J ( α q ) · ξ = h α q , ξ Q ( q ) i , (28.2) D. D. Holm Imp erial College London Applications of P oisson Geometry 112 where α q ∈ T ∗ q Q and ξ ∈ g , where ξ Q is the inﬁnitesimal generator of the action of G on Q asso ciated to the Lie algebra element ξ , and where h α q , ξ Q ( q ) i is the natural pairing of an element of T ∗ q Q with an elemen t of T q Q . No w w e apply this formula to the sp ecial case in which the group G is the diﬀeomorphism group Diﬀ ( R n ), the manifold Q is Emb( S, R n ) and where the action of the group on Em b( S, R n ) is giv en b y (28.1). The sense in which the Lie algebra of G = Diﬀ is the space g = X of vector ﬁelds is well- understo od. Hence, its dual is naturally regarded as the space of one-form densities. The momentum map is th us a map J : T ∗ Em b( S, R n ) → X ∗ . With J given by (28.2), we only need to w ork out this form ula. First, we shall work out the inﬁnitesimal generators. Let X ∈ X b e a Lie algebra elemen t. By diﬀerentiating the action (28.1) with resp ect to η in the direction of X at the iden tit y elemen t we ﬁnd that the inﬁnitesimal generator is giv en by X Emb( S , R n ) ( Q ) = X ◦ Q . Th us, taking α q to b e the cotangen t v ector ( Q , P ), equation (28.2) giv es h J ( Q , P ) , X i = h ( Q , P ) , X ◦ Q i = Z S P i ( s ) X i ( Q ( s )) d k s. On the other hand, note that the righ t hand side of (27.8) (again with the index a suppressed, and with t suppressed as w ell), when paired with the Lie algebra element X is  Z S P ( s ) δ ( x − Q ( s )) d k s, X  = Z R n Z S  P i ( s ) δ ( x − Q ( s )) d k s  X i ( x ) d n x = Z S P i ( s ) X i ( Q ( s ) d k s. This sho ws that the expression given b y (27.8) is equal to J and so the result is prov ed.  Second Pro of. As is standard (see, for example, MaRa1994), one can c haracterize momentum maps b y means of the follo wing relation, required to hold for all functions F on T ∗ Em b( S, R n ); that is, functions of Q and P : { F , h J , ξ i} = ξ P [ F ] . (28.3) In our case, we shall take J to b e given by the solution ansatz and verify that it satisﬁes this relation. T o do so, let ξ ∈ X so that the left side of (28.3) b ecomes  F , Z S P i ( s ) ξ i ( Q ( s )) d k s  = Z S  δ F δ Q i ξ i ( Q ( s )) − P i ( s ) δ F δ P j δ δ Q j ξ i ( Q ( s ))  d k s . On the other hand, one can directly compute from the deﬁnitions that the inﬁnitesimal generator of the action on the space T ∗ Em b( S, R n ) corresp onding to the vector ﬁeld ξ i ( x ) ∂ ∂ Q i (a Lie algebra elemen t), is given b y (see MaRa1994, form ula (12.1.14)): δ Q = ξ ◦ Q , δ P = − P i ( s ) ∂ ∂ Q ξ i ( Q ( s )) , whic h veriﬁes that (28.3) holds. An important elemen t left out in this pro of so far is that it do es not mak e clear that the momentum map is e quivariant , a condition needed for the momen tum map to b e P oisson. The ﬁrst proof to ok D. D. Holm Imp erial College London Applications of P oisson Geometry 113 care of this automatically since momentum maps for c otangent lifte d actions ar e always e quivariant and hence are Poisson. Th us, to complete the second proof, we need to c heck directly that the momentum map is equiv ari- an t. Actually , w e shall only chec k that it is inﬁnitesimally inv ariant b y sho wing that it is a P oisson map from T ∗ Em b( S, R n ) to the space of m ’s (the dual of the Lie algebra of X ) with its Lie-Poisson brac k et. This sort of approach to c haracterize equiv ariant momentum maps is discussed in an interesting w a y in [W e2002]. The follo wing direct computation sho ws that the singular solution momentum map (27.8) is Pois- son. This is accomplished b y using the canonical Poisson brack ets for { P } , { Q } and applying the c hain rule to compute  m i ( x ) , m j ( y )  , with notation δ 0 k ( y ) ≡ ∂ δ ( y ) /∂ y k . W e get  m i ( x ) , m j ( y )  =  N X a =1 Z ds P a i ( s, t ) δ ( x − Q a ( s, t )) , N X b =1 Z ds 0 P b j ( s 0 , t ) δ ( y − Q b ( s 0 , t ))  = N X a,b =1 Z Z dsds 0  { P a i ( s ) , P b j ( s 0 ) } δ ( x − Q a ( s )) δ ( y − Q b ( s 0 )) − { P a i ( s ) , Q b k ( s 0 ) } P b j ( s 0 ) δ ( x − Q a ( s )) δ 0 k ( y − Q b ( s 0 )) − { Q a k ( s ) , P b j ( s 0 ) } P a i ( s ) δ 0 k ( x − Q a ( s )) δ ( y − Q b ( s 0 )) + { Q a k ( s ) , Q b ` ( s 0 ) } P a i ( s ) P b j ( s 0 ) δ 0 k ( x − Q a ( s )) δ 0 ` ( y − Q b ( s 0 ))  . Substituting the canonical P oisson brack et relations { P a i ( s ) , P b j ( s 0 ) } = 0 { Q a k ( s ) , Q b ` ( s 0 ) } = 0 , and { Q a k ( s ) , P b j ( s 0 ) } = δ ab δ kj δ ( s − s 0 ) in to the preceding computation yields,  m i ( x ) , m j ( y )  =  N X a =1 Z dsP a i ( s, t ) δ ( x − Q a ( s, t )) , N X b =1 Z ds 0 P b j ( s 0 , t ) δ ( y − Q b ( s 0 , t ))  = N X a =1 Z dsP a j ( s ) δ ( x − Q a ( s )) δ 0 i ( y − Q a ( s )) − N X a =1 Z dsP a i ( s ) δ 0 j ( x − Q a ( s )) δ ( y − Q a ( s )) = −  m j ( x ) ∂ ∂ x i + ∂ ∂ x j m i ( x )  δ ( x − y ) . Th us,  m i ( x ) , m j ( y )  = −  m j ( x ) ∂ ∂ x i + ∂ ∂ x j m i ( x )  δ ( x − y ) , (28.4) D. D. Holm Imp erial College London Applications of P oisson Geometry 114 whic h is readily chec ked to b e the Lie-Poisson brack et on the space of m ’s, restricted to their singular supp ort. This completes the second pro of of theorem.  Eac h of these pro ofs has sho wn the following basic fact. Corollary 28.2 The singular solution momentum map deﬁne d by the singular solution ansatz (27.8) , namely, J Sing : T ∗ Em b( S, R n ) → X ( R n ) ∗ is a Poisson map fr om the c anonic al Poisson structur e on T ∗ Em b( S, R n ) to the Lie-Poisson structur e on X ( R n ) ∗ . This is p erhaps the most basic property of the singular solution momen tum map. Some of its more sophisticated prop erties are outlined in [HoMa2004]. Pulling Back the Equations. Since the solution ansatz (27.8) has b een sho wn in the preceding Corollary to b e a P oisson map, the pull back of the Hamiltonian from X ∗ to T ∗ Em b( S, R n ) giv es equations of motion on the latter space that pro ject to the equations on X ∗ . Thus, the b asic fact that the momentum map J Sing is Poisson explains why the functions Q a ( s, t ) and P a ( s, t ) satisfy c anonic al Hamiltonian e quations. Note that the coordinate s ∈ R k that lab els these functions is a “Lagrangian coordinate” in the sense that it do es not ev olv e in time but rather lab els the solution. In terms of the pairing h· , ·i : g ∗ × g → R , (28.5) b et w een the Lie algebra g (vector ﬁelds in R n ) and its dual g ∗ (one-form densities in R n ), the following relation holds for measure-v alued solutions under the momentum map (27.8), h m , u i = Z m · u d n x , L 2 pairing for m & u ∈ R n , = Z Z N X a , b =1  P a ( s, t ) · P b ( s 0 , t )  G  Q a ( s, t ) − Q b ( s 0 , t )  ds ds 0 = Z N X a =1 P a ( s, t ) · ∂ Q a ( s, t ) ∂ t ds ≡ h h P , ˙ Q i i , (28.6) whic h is the natural pairing b et w een the p oints ( Q , P ) ∈ T ∗ Em b( S, R n ) and ( Q , ˙ Q ) ∈ T Em b( S, R n ). This corresp onds to preserv ation of the action of the Lagrangian ` [ u ] under cotangent lift of Diﬀ ( R n ). The pull-bac k of the Hamiltonian H [ m ] deﬁned on the dual of the Lie algebra g ∗ , to T ∗ Em b( S, R n ) is easily seen to b e consisten t with what w e had b efore: H [ m ] ≡ 1 2 h m , G ∗ m i = 1 2 h h P , G ∗ P i i ≡ H N [ P , Q ] . (28.7) In summary , in concert with the Poisson nature of the singular solution momentum map, w e see that the singular solutions in terms of Q and P satisfy Hamiltonian equations and also deﬁne an in v ariant solution set for the EPDiﬀ equations. In fact, This invariant solution set is a sp e cial c o adjoint orbit for the diﬀe omorphism gr oup, as we shal l discuss in the next se ction. D. D. Holm Imp erial College London Applications of P oisson Geometry 115 29 The Geometry of the Momentum Map In this section we explore the geometry of the singular solution momentum map discussed earlier in a little more detail. The treatmen t is formal, in the sense that there are a n umber of technical issues in the inﬁnite dimensional case that will b e left open. W e will mention a few of these as w e pro ceed. 29.1 Coadjoin t Orbits. W e claim that the image of the singular solution momentum map is a c o adjoint orbit in X ∗ . This means that (mo dulo some issues of connectedness and smo othness, whic h we do not consider here) the solution ansatz given by (27.8) deﬁnes a coadjoint orbit in the space of all one-form densities, regarded as the dual of the Lie algebra of the diﬀeomorphism group. These coadjoin t orbits should b e though t of as singular orbits—that is, due to their sp ecial nature, they are not generic. Recognizing them as coadjoint orbits is one w a y of gaining further insight in to wh y the singular solutions form dynamically inv arian t sets—it is a general fact that coadjoint orbits in g ∗ are symple ctic submanifolds of the Lie-P oisson manifold g ∗ (in our case X ( R n ) ∗ ) and, correspondingly , are dynamically in v ariant for an y Hamiltonian system on g ∗ . The idea of the proof of our claim is simply this: whenever one has an equiv ariant momentum map J : P → g ∗ for the action of a group G on a symplectic or P oisson manifold P , and that action is transitiv e, then the image of J is an orbit (or at least a piece of an orbit). This general result, due to Kostan t, is stated more precisely in [MaRa1994], Theorem 14.4.5. Roughly sp eaking, the reason that transitivit y holds in our case is b ecause one can “mov e the images of the manifolds S around at will with arbitrary v elo cit y ﬁelds” using diﬀeomorphisms of R n . 29.2 The Momentum map J S and the Kelvin circulation theorem. The momentum map J Sing in volv es Diﬀ ( R n ), the left action of the diﬀeomorphism group on the space of em b eddings Em b( S, R n ) b y smo oth maps of the target space R n , namely , Diﬀ ( R n ) : Q · η = η ◦ Q , (29.1) where, recall, Q : S → R n . As ab o ve, the cotangent bundle T ∗ Em b( S, R n ) is identiﬁed with the space of pairs of maps ( Q , P ), with Q : S → R n and P : S → T ∗ R n . Ho wev er, there is another momentum map J S asso ciated with the right action of the diﬀeomor- phism group of S on the embeddings Emb( S, R n ) by smo oth maps of the “Lagrangian lab els” S (ﬂuid particle relab eling b y η : S → S ). This action is given by Diﬀ ( S ) : Q · η = Q ◦ η . (29.2) The inﬁnitesimal generator of this righ t action is X Emb( S , R n ) ( Q ) = d dt    t =0 Q ◦ η t = T Q ◦ X. (29.3) where X ∈ X is tangen t to the curve η t at t = 0. Th us, again taking N = 1 (so w e suppress the index a ) and also letting α q in the momen tum map form ula (28.2) b e the cotangen t vector ( Q , P ), D. D. Holm Imp erial College London Applications of P oisson Geometry 116 one computes J S : h J S ( Q , P ) , X i = h ( Q , P ) , T Q · X i = Z S P i ( s ) ∂ Q i ( s ) ∂ s m X m ( s ) d k s = Z S X  P ( s ) · d Q ( s )  d k s =  Z S P ( s ) · d Q ( s ) ⊗ d k s , X ( s )  = h P · d Q , X i . Consequen tly , the momentum map form ula (28.2) yields J S ( Q , P ) = P · d Q , (29.4) with the indicated pairing of the one-form densit y P · d Q with the v ector ﬁeld X . W e ha ve set things up so that the follo wing is true. Prop osition 29.1 The momentum map J S is pr eserve d by the evolution e quations (27.10-27.11) for Q and P . Pro of. It is enough to notice that the Hamiltonian H N in equation (27.12) is inv ariant under the cotangen t lift of the action of Diﬀ ( S ); it merely amounts to the in v ariance of the integral o ver S under reparametrization; that is, the c hange of v ariables formula; keep in mind that P includes a densit y factor. Remark 29.2 • This r esult is similar to the Kelvin-No ether the or em for cir culation Γ of an ide al ﬂuid, which may b e written as Γ = H c ( s ) D ( s ) − 1 P ( s ) · d Q ( s ) for e ach L agr angian cir cuit c ( s ) , wher e D is the mass density and P is again the c anonic al momentum density. This similarity should c ome as no surprise, b e c ause the Kelvin-No ether the or em for ide al ﬂuids arises fr om invarianc e of Hamilton ’s principle under ﬂuid p ar c el r elab eling by the same right action of the diﬀe omorphism gr oup, as in (29.2). • Note that, b eing an e quivariant momentum map, the map J S , as with J Sing , is also a Poisson map. That is, substituting the c anonic al Poisson br acket into r elation (29.4) ; that is, the r elation M ( x ) = P i P i ( x ) ∇ Q i ( x ) yields the Lie-Poisson br acket on the sp ac e of M ’s. We use the diﬀer ent notations m and M b e c ause these quantities ar e analo gous to the b o dy and sp atial angular momentum for rigid b o dy me chanics. In fact, the quantity m given by the solution A nsatz; sp e ciﬁc al ly, m = J Sing ( Q , P ) gives the singular solutions of the EPDiﬀ e quations, while M ( x ) = J S ( Q , P ) = P i P i ( x ) ∇ Q i ( x ) is a c onserve d quantity. • In the language of ﬂuid me chanics, the expr ession of m in terms of ( Q , P ) is an example of a Clebsch r epr esentation , which expr esses the solution of the EPDiﬀ e quations in terms of c anonic al variables that evolve by standar d c anonic al Hamilton e quations. This has b e en known in the c ase of ﬂuid me chanics for mor e than 100 ye ars. F or mo dern discussions of the Clebsch r epr esentation for ide al ﬂuids, se e, for example, [HoKu1983, MaWe1983]. D. D. Holm Imp erial College London Applications of P oisson Geometry 117 • One mor e r emark is in or der; namely the sp e cial c ase in which S = M is of c ourse al lowe d. In this c ase, Q c orr esp onds to the map η itself and P just c orr esp onds to its c onjugate momentum. The quantity m c orr esp onds to the sp atial (dynamic) momentum density (that is, right tr anslation of P to the identity), while M c orr esp onds to the c onserve d “b o dy” momentum density (that is, left tr anslation of P to the identity). 29.3 Brief summary Em b( S, R n ) admits t wo group actions. These are: the group Diﬀ ( S ) of diﬀeomorphisms of S , whic h acts b y comp osition on the right ; and the group Diﬀ ( R n ) whic h acts by composition on the left . The group Diﬀ ( R n ) acting from the left produces the singular solution momen tum map, J Sing . The action of Diﬀ ( S ) from the righ t pro duces the conserved momen tum map J S : T ∗ Em b( S, R n ) → X ( S ) ∗ . W e no w assemble both momentum maps in to one ﬁgure as follo ws: T ∗ Em b( S, M ) J Sing J S X ( M ) ∗ X ( S ) ∗      @ @ @ @ R 30 The Euler-P oincar´ e framew ork: ﬂuids ` a la [HoMaRa1998a] Almost all ﬂuid models of interest admit the following general assumptions. These assumptions form the basis of the Euler-Poincar ´ e theorem for Contin uua that we shall state later in this section, after in tro ducing the notation necessary for dealing geometrically with the reduction of Hamilton’s Principle from the material (or Lagrangian) picture of ﬂuid dynamics, to the spatial (or Eulerian) picture. This theorem w as ﬁrst stated and prov ed in [HoMaRa1998a], to whic h w e refer for additional details, as w ell as for abstract deﬁnitions and pro ofs. Basic assumptions underlying the E uler-P oincar´ e theorem for con tinua • There is a right representation of a Lie group G on the vector space V and G acts in the natural w ay on the right on T G × V ∗ : ( U g , a ) h = ( U g h, ah ). • The Lagrangian function L : T G × V ∗ → R is righ t G –inv ariant under the isotrop y group of a 0 ∈ V ∗ . 13 • In particular, if a 0 ∈ V ∗ , deﬁne the Lagrangian L a 0 : T G → R b y L a 0 ( U g ) = L ( U g , a 0 ). Then L a 0 is right in v ariant under the lift to T G of the right action of G a 0 on G , where G a 0 is the isotrop y group of a 0 . • Righ t G –in v ariance of L p ermits one to deﬁne the Lagrangian on the Lie algebra g of the group G . Namely , ` : g × V ∗ → R is deﬁned by , ` ( u, a ) = L  U g g − 1 ( t ) , a 0 g − 1 ( t )  = L ( U g , a 0 ) , 13 F or ﬂuid dynamics, right G –inv ariance of the Lagrangian function L is traditionally called “particle relabeling symmetry .” D. D. Holm Imp erial College London Applications of P oisson Geometry 118 where u = U g g − 1 ( t ) and a = a 0 g − 1 ( t ) . Conv ersely , this relation deﬁnes for any ` : g × V ∗ → R a righ t G –inv ariant function L : T G × V ∗ → R . • F or a curve g ( t ) ∈ G, let u ( t ) := ˙ g ( t ) g ( t ) − 1 and deﬁne the curv e a ( t ) as the unique solution of the linear diﬀerential equation with time dep endent co eﬃcien ts ˙ a ( t ) = − a ( t ) u ( t ), where the action of an elemen t of the Lie algebra u ∈ g on an adv ected quantit y a ∈ V ∗ is denoted by concatenation from the righ t. The solution with initial condition a (0) = a 0 ∈ V ∗ can b e written as a ( t ) = a 0 g ( t ) − 1 . Notation for reduction of Hamil ton’s Principle b y symmetries • Let g ( D ) denote the space of vector ﬁelds on D of some ﬁxed diﬀerentiabilit y class. These vector ﬁelds are endo wed with the Lie br acket giv en in comp onents b y (summing on rep eated indices) [ u , v ] i = u j ∂ v i ∂ x j − v j ∂ u i ∂ x j . (30.1) The notation ad u v := [ u , v ] formally denotes the adjoin t action of the right Lie algebra of Diﬀ ( D ) on itself. • Iden tify the Lie algebra of vector ﬁelds g with its dual g ∗ b y using the L 2 pairing h u , v i = Z D u · v dV . (30.2) • Let g ( D ) ∗ denote the geometric dual space of g ( D ), that is, g ( D ) ∗ := Λ 1 ( D ) ⊗ Den( D ). This is the space of one–form densities on D . If m ⊗ dV ∈ Λ 1 ( D ) ⊗ Den( D ), then the pairing of m ⊗ dV with u ∈ g ( D ) is given b y the L 2 pairing, h m ⊗ dV , u i = Z D m · u dV (30.3) where m · u is the standard con traction of a one–form m with a vector ﬁeld u . • F or u ∈ g ( D ) and m ⊗ dV ∈ g ( D ) ∗ , the dual of the adjoin t representation is deﬁned b y h ad ∗ u ( m ⊗ dV ) , v i = − Z D m · ad u v dV = − Z D m · [ u , v ] dV (30.4) and its expression is ad ∗ u ( m ⊗ dV ) = ( £ u m + (div dV u ) m ) ⊗ dV = £ u ( m ⊗ dV ) , (30.5) where div dV u is the divergence of u relative to the measure dV , that is, £ u dV = (div dV u ) dV . Hence, ad ∗ u coincides with the Lie-deriv ative £ u for one-form densities. • If u = u j ∂ /∂ x j , m = m i dx i , then the one–form factor in the preceding formula for ad ∗ u ( m ⊗ dV ) has the c o or dinate expr ession  ad ∗ u m  i dx i =  u j ∂ m i ∂ x j + m j ∂ u j ∂ x i + (div dV u ) m i  dx i (30.6) =  ∂ ∂ x j ( u j m i ) + m j ∂ u j ∂ x i  dx i . (30.7) The last equality assumes that the divergence is taken relativ e to the standard measure dV = d n x in R n . (On a Riemannian manifold the metric divergence needs to be used.) D. D. Holm Imp erial College London Applications of P oisson Geometry 119 Con v entions and terminology in contin uum mec hanics Throughout the rest of the lecture notes, we shall follow [HoMaRa1998a] in using the conv en tions and terminology for the standard quan tities in contin uum mec hanics. Deﬁnition 30.1 Elements of D r epr esenting the material p articles of the system ar e denote d by X ; their c o or dinates X A , A = 1 , ..., n may thus b e r e gar de d as the p article lab els . • A c onﬁgur ation , which we typic al ly denote by η , or g , is an element of Diﬀ ( D ) . • A motion , denote d as η t or alternatively as g ( t ) , is a time dep endent curve in Diﬀ ( D ) . Deﬁnition 30.2 The L agr angian , or material velo city U ( X, t ) of the c ontinuum along the motion η t or g ( t ) is deﬁne d by taking the time derivative of the motion ke eping the p article lab els X ﬁxe d: U ( X , t ) := dη t ( X ) dt := ∂ ∂ t     X η t ( X ) := ˙ g ( t ) · X . These ar e c onvenient shorthand notations for the time derivative at ﬁxe d L agr angian c o or dinate X . Consistent with this deﬁnition of material velo city, the tangent sp ac e to Diﬀ ( D ) at η ∈ Diﬀ ( D ) is given by T η Diﬀ ( D ) = { U η : D → T D | U η ( X ) ∈ T η ( X ) D } . Elements of T η Diﬀ ( D ) ar e usual ly thought of as ve ctor ﬁelds on D c overing η . The tangent lift of right tr anslations on T Diﬀ ( D ) by ϕ ∈ Diﬀ ( D ) is given by U η ϕ := T η R ϕ ( U η ) = U η ◦ ϕ . Deﬁnition 30.3 During a motion η t or g ( t ) , the p article lab ele d by X describ es a p ath in D , whose p oints x ( X , t ) := η t ( X ) := g ( t ) · X , ar e c al le d the Eulerian or sp atial p oints of this p ath, which is also c al le d the L agr angian tr aje c- tory , b e c ause a L a gr angian ﬂuid p ar c el fol lows this p ath in sp ac e. The derivative u ( x, t ) of this p ath, evaluate d at ﬁxe d Eulerian p oint x , is c al le d the Eulerian or sp atial velo city of the system: u ( x, t ) := u ( η t ( X ) , t ) := U ( X , t ) := ∂ ∂ t     X η t ( X ) := ˙ g ( t ) · X := ˙ g ( t ) g − 1 ( t ) · x . Thus the Eulerian velo city u is a time dep endent ve ctor ﬁeld on D , denote d as u t ∈ g ( D ) , wher e u t ( x ) := u ( x, t ) . We also have the fundamental r elationships U t = u t ◦ η t and u t = ˙ g ( t ) g − 1 ( t ) , wher e we denote U t ( X ) := U ( X , t ) . Deﬁnition 30.4 The r epr esentation sp ac e V ∗ of Diﬀ ( D ) in c ontinuum me chanics is often some subsp ac e of the tensor ﬁeld densities on D , denote d as T ( D ) ⊗ Den( D ) , and the r epr esentation is given by pul l b ack. It is thus a right r epr esentation of Diﬀ ( D ) on T ( D ) ⊗ Den( D ) . The right action of the Lie algebr a g ( D ) on V ∗ is denote d as c onc atenation fr om the right . That is, we denote a u := £ u a , which is the Lie derivative of the tensor ﬁeld density a along the ve ctor ﬁeld u . D. D. Holm Imp erial College London Applications of P oisson Geometry 120 Deﬁnition 30.5 The L agr angian of a c ontinuum me chanic al system is a function L : T Diﬀ ( D ) × V ∗ → R , which is right invariant r elative to the tangent lift of right tr anslation of Diﬀ ( D ) on itself and pul l b ack on the tensor ﬁeld densities. Invarianc e of the L agr angian L induc es a function ` : g ( D ) × V ∗ → R given by ` ( u , a ) = L ( u ◦ η , η ∗ a ) = L ( U , a 0 ) , wher e u ∈ g ( D ) and a ∈ V ∗ ⊂ T ( D ) ⊗ Den( D ) , and wher e η ∗ a denotes the pul l b ack of a by the diﬀe omorphism η and u is the Eulerian velo city. That is, U = u ◦ η and a 0 = η ∗ a . (30.8) The evolution of a is by right action, given by the e quation ˙ a = − £ u a = − a u . (30.9) The solution of this e quation, for the initial c ondition a 0 , is a ( t ) = η t ∗ a 0 = a 0 g − 1 ( t ) , (30.10) wher e the lower star denotes the push forwar d op er ation and η t is the ﬂow of u = ˙ g g − 1 ( t ) . Deﬁnition 30.6 A dve cte d Eulerian quantities ar e deﬁne d in c ontinuum me chanics to b e those variables which ar e Lie tr ansp orte d by the ﬂow of the Eulerian velo city ﬁeld. Using this standar d terminolo gy, e quation (30.9), or its solution (30.10) states that the tensor ﬁeld density a ( t ) (which may include mass density and other Eulerian quantities) is adve cte d. Remark 30.7 (Dual tensors) As we mentione d, typic al ly V ∗ ⊂ T ( D ) ⊗ Den( D ) for c ontinuum me chanics. On a gener al manifold, tensors of a given typ e have natur al duals. F or example, symmetric c ovariant tensors ar e dual to symmetric c ontr avariant tensor densities, the p airing b eing given by the inte gr ation of the natur al c ontr action of these tensors. Likewise, k –forms ar e natur al ly dual to ( n − k ) – forms, the p airing b eing given by taking the inte gr al of their we dge pr o duct. Deﬁnition 30.8 The diamond op er ation  b etwe en elements of V and V ∗ pr o duc es an element of the dual Lie algebr a g ( D ) ∗ and is deﬁne d as h b  a, w i = − Z D b · £ w a , (30.11) wher e b · £ w a denotes the c ontr action, as describ e d ab ove, of elements of V and elements of V ∗ and w ∈ g ( D ) . (These op er ations do not dep end on a Riemannian structur e.) F or a path η t ∈ Diﬀ ( D ), let u ( x, t ) b e its Eulerian velocity and consider the curve a ( t ) with initial condition a 0 giv en by the equation ˙ a + £ u a = 0 . (30.12) Let the Lagrangian L a 0 ( U ) := L ( U , a 0 ) b e right-in v arian t under Diﬀ ( D ). W e can no w state the Euler–P oincar´ e Theorem for Contin ua of [HoMaRa1998a]. Theorem 30.9 (Euler–Poincar ´ e Theorem for Contin ua.) Consider a p ath η t in Diﬀ ( D ) with L agr angian velo city U and Eulerian velo city u . The fol lowing ar e e quivalent: D. D. Holm Imp erial College London Applications of P oisson Geometry 121 i Hamilton ’s variational principle δ Z t 2 t 1 L ( X , U t ( X ) , a 0 ( X )) dt = 0 (30.13) holds, for variations δ η t vanishing at the endp oints. ii η t satisﬁes the Euler–L agr ange e quations for L a 0 on Diﬀ ( D ) . iii The c onstr aine d variational principle in Eulerian c o or dinates δ Z t 2 t 1 ` ( u , a ) dt = 0 (30.14) holds on g ( D ) × V ∗ , using variations of the form δ u = ∂ w ∂ t + [ u , w ] = ∂ w ∂ t + ad u w , δ a = − £ w a, (30.15) wher e w t = δ η t ◦ η − 1 t vanishes at the endp oints. iv The Euler–Poinc ar´ e e quations for c ontinua ∂ ∂ t δ ` δ u = − ad ∗ u δ ` δ u + δ ` δ a  a = − £ u δ ` δ u + δ ` δ a  a , (30.16) hold, with auxiliary e quations ( ∂ t + £ u ) a = 0 for e ach adve cte d quantity a ( t ) . The  op er ation deﬁne d in (30.11) ne e ds to b e determine d on a c ase by c ase b asis, dep ending on the natur e of the tensor a ( t ) . The variation m = δ `/δ u is a one–form density and we have use d r elation (30.5) in the last step of e quation (30.16). W e refer to [HoMaRa1998a] for the pro of of this theorem in the abstract setting. W e shall see some of the features of this result in the concrete setting of con tinuum mec hanics shortly . Discussion of the Euler-P oincar´ e equations The follo wing string of equalities shows dir e ctly that iii is equiv alen t to iv : 0 = δ Z t 2 t 1 l ( u , a ) dt = Z t 2 t 1  δ l δ u · δ u + δ l δ a · δ a  dt = Z t 2 t 1  δ l δ u ·  ∂ w ∂ t − ad u w  − δ l δ a · £ w a  dt = Z t 2 t 1 w ·  − ∂ ∂ t δ l δ u − ad ∗ u δ l δ u + δ l δ a  a  dt . (30.17) The rest of the pro of follows essen tially the same track as the proof of the pure Euler-Poincar ´ e theorem, mo dulo sligh t c hanges to accomo date the adv ected quantities. In the absence of dissipation, most Eulerian ﬂuid equations 14 can b e written in the EP form in equation (30.16), ∂ ∂ t δ ` δ u + ad ∗ u δ ` δ u = δ ` δ a  a , with  ∂ t + £ u  a = 0 . (30.18) 14 Exceptions to this statemen t are certain multiphase ﬂuids, and complex ﬂuids with active in ternal degrees of freedom suc h as liquid crystals. These require a further extension, not discussed here. D. D. Holm Imp erial College London Applications of P oisson Geometry 122 Equation (30.18) is Newton ’s L aw : The Eulerian time deriv ative of the momen tum densit y m = δ `/δ u (a one-form density dual to the velocity u ) is equal to the force densit y ( δ `/δ a )  a , with the  op eration deﬁned in (30.11). Thus, Newton’s La w is written in the Eulerian ﬂuid represen tation as, 15 d dt    Lag m :=  ∂ t + £ u  m = δ ` δ a  a , with d dt    Lag a :=  ∂ t + £ u  a = 0 . (30.19) • The left side of the EP equation in (30.19) describes the ﬂuid’s dynamics due to its kinetic energy . A ﬂuid’s kinetic energy typically deﬁnes a norm for the Eulerian ﬂuid v elo cit y , K E = 1 2 k u k 2 . The left side of the EP equation is the ge o desic part of its evolution, with resp ect to this norm. See [ArKh1998] for discussions of this in terpretation of ideal incompressible ﬂo w and references to the literature. How ev er, in a gravitational ﬁeld, for example, there will also b e dynamics due to p oten tial energy . And this dynamics will b y gov erned b y the right side of the EP equation. • The right side of the EP equation in (30.19) mo diﬁes the geo desic motion. Naturally , the right side of the EP equation is also a geometrical quantit y . The diamond op eration  represents the dual of the Lie algebra action of v ectors ﬁelds on the tensor a . Here δ `/δa is the dual tensor, under the natural pairing (usually , L 2 pairing) h · , · i that is induced b y the v ariational deriv ative of the Lagrangian ` ( u , a ). The diamond op eration  is deﬁned in terms of this pairing in (30.11). F or the L 2 pairing, this is in tegration by parts of (min us) the Lie deriv ative in (30.11). • The quantit y a is typically a tensor (e.g., a density , a scalar, or a diﬀeren tial form) and w e shall sum ov er the v arious t yp es of tensors a that are inv olv ed in the ﬂuid description. The second equation in (30.19) states that eac h tensor a is carried along b y the Eulerian ﬂuid velocity u . Th us, a is for ﬂuid “attribute,” and its Eulerian evolution is given by minus its Lie deriv ative, − £ u a . That is, a stands for the set of ﬂuid attributes that eac h Lagrangian ﬂuid parcel carries around (advects), such as its buoy ancy , which is determined by its individual salt, or heat con tent, in ocean circulation. • Man y examples of ho w equation (30.19) arises in the dynamics of contin uous media are given in [HoMaRa1998a]. The EP form of the Eulerian ﬂuid description in (30.19) is analogous to the classical dynamics of rigid bo dies (and tops, under gra vity) in b o dy co ordinates. Rigid bo dies and tops are also gov erned by Euler-P oincar´ e equations, as Poincar ´ e show ed in a tw o-page pap er with no references, ov er a century ago [Po1901]. F or mo dern discussions of the EP theory , see, e.g., [MaRa1994], or [HoMaRa1998a]. Exercise 30.10 F or what typ es of tensors a 0 c an one r e c ast the EP e quations for c ontinua (30.16) as ge o desic motion, by using a version of the Kaluza-Klein c onstruction? 15 In coordinates, a one-form density tak es the form m · d x ⊗ dV and the EP equation (30.16) is giv en neumonically b y d dt ˛ ˛ ˛ Lag ` m · d x ⊗ dV ´ = d m dt ˛ ˛ ˛ Lag · d x ⊗ dV | {z } Adv ection + m · d u ⊗ dV | {z } Stretc hing + m · d x ⊗ ( ∇ · u ) dV | {z } Expansion = δ ` δ a  a with d dt ˛ ˛ ˛ Lag d x := ` ∂ t + £ u ´ d x = d u = u ,j dx j , upon using commutation of Lie deriv ativ e and exterior deriv ative. Compare this formula with the deﬁnition of ad ∗ u ( m ⊗ dV ) in equation (30.6). D. D. Holm Imp erial College London Applications of P oisson Geometry 123 30.1 Corollary of the EP theorem: the Kelvin-No ether circulation theorem Corollary 30.11 (Kelvin-No ether Circulation Theorem.) Assume u ( x, t ) satisﬁes the Euler– Poinc ar´ e e quations for c ontinua: ∂ ∂ t  δ ` δ u  = − £ u  δ ` δ u  + δ ` δ a  a and the quantity a satisﬁes the adve ction r elation ∂ a ∂ t + £ u a = 0 . (30.20) L et η t b e the ﬂow of the Eulerian velo city ﬁeld u , that is, u = ( dη t /dt ) ◦ η − 1 t . Deﬁne the adve cte d ﬂuid lo op γ t := η t ◦ γ 0 and the cir culation map I ( t ) by I ( t ) = I γ t 1 D δ ` δ u . (30.21) In the cir culation map I ( t ) the adve cte d mass density D t satisﬁes the push forwar d r elation D t = η ∗ D 0 . This implies the adve ction r elation (30.20) with a = D , namely, the c ontinuity e quation, ∂ t D + div D u = 0 . Then the map I ( t ) satisﬁes the Kelvin cir culation r elation , d dt I ( t ) = I γ t 1 D δ ` δ a  a . (30.22) Both an abstract pro of of the Kelvin-No ether Circulation Theorem and a pro of tailored for the case of con tinuum mec hanical systems are given in [HoMaRa1998a]. W e provide a version of the latter b elo w. Pro of. First we change v ariables in the expression for I ( t ): I ( t ) = I γ t 1 D t δ l δ u = I γ 0 η ∗ t  1 D t δ l δ u  = I γ 0 1 D 0 η ∗ t  δ l δ u  . Next, w e use the Lie deriv ativ e formula, namely d dt ( η ∗ t α t ) = η ∗ t  ∂ ∂ t α t + £ u α t  , applied to a one–form densit y α t . This formula giv es d dt I ( t ) = d dt I γ 0 1 D 0 η ∗ t  δ l δ u  = I γ 0 1 D 0 d dt  η ∗ t  δ l δ u  = I γ 0 1 D 0 η ∗ t  ∂ ∂ t  δ l δ u  + £ u  δ l δ u  . By the Euler–P oincar´ e equations (30.16), this b ecomes d dt I ( t ) = I γ 0 1 D 0 η ∗ t  δ l δ a  a  = I γ t 1 D t  δ l δ a  a  , again b y the change of v ariables formula. D. D. Holm Imp erial College London Applications of P oisson Geometry 124 Corollary 30.12 Sinc e the last expr ession holds for every lo op γ t , we may write it as  ∂ ∂ t + £ u  1 D δ l δ u = 1 D δ l δ a  a . (30.23) Remark 30.13 The Kelvin-No ether the or em is c al le d so her e b e c ause its derivation r elies on the invarianc e of the L agr angian L under the p article r elab eling symmetry, and No ether’s the or em is asso ciate d with this symmetry. However, the r esult (30.22) is the Kelvin cir culation the or em : the cir culation inte gr al I ( t ) ar ound any ﬂuid lo op ( γ t , moving with the velo city of the ﬂuid p ar c els u ) is invariant under the ﬂuid motion. These two statements ar e e quivalent. We note that two velo cities app e ar in the inte gr and I ( t ) : the ﬂuid velo city u and D − 1 δ `/δ u . The latter velo city is the momentum density m = δ `/δ u divide d by the mass density D . These two velo cities ar e the b asic ingr e dients for p erforming mo deling and analysis in any ide al ﬂuid pr oblem. One simply ne e ds to put these ingr e dients to gether in the Euler-Poinc ar´ e the or em and its c or ol lary, the Kelvin-No ether the or em. 31 Euler–P oincar ´ e theorem & GFD (geophysical ﬂuid dynamics) 31.1 V ariational F ormulae in Three Dimensions W e compute explicit formulae for the v ariations δ a in the cases that the set of tensors a is drawn from a set of scalar ﬁelds and densities on R 3 . W e s hall denote this symbolically by writing a ∈ { b, D d 3 x } . (31.1) W e hav e seen that inv ariance of the set a in the Lagrangian picture under the dynamics of u implies in the Eulerian picture that  ∂ ∂ t + £ u  a = 0 , where £ u denotes Lie deriv ative with resp ect to the velocity v ector ﬁ eld u . Hence, for a ﬂuid dynamical Eulerian action S = R dt ` ( u ; b, D ), the adv ected v ariables b and D satisfy the following Lie-deriv ative relations,  ∂ ∂ t + £ u  b = 0 , or ∂ b ∂ t = − u · ∇ b , (31.2)  ∂ ∂ t + £ u  D d 3 x = 0 , or ∂ D ∂ t = − ∇ · ( D u ) . (31.3) In ﬂuid dynamical applications, the advected Eulerian v ariables b and D d 3 x represent the buoy ancy b (or sp eciﬁc entrop y , for the compressible case) and volume eleme n t (or mass densit y) D d 3 x , resp ec- tiv ely . According to Theorem 30.9, equation (30.14), the v ariations of the tensor functions a at ﬁxed x and t are also given b y Lie deriv atives, namely δ a = − £ w a , or δ b = − £ w b = − w · ∇ b , δ D d 3 x = − £ w ( D d 3 x ) = − ∇ · ( D w ) d 3 x . (31.4) D. D. Holm Imp erial College London Applications of P oisson Geometry 125 Hence, Hamilton’s principle (30.14) with this dep endence yields 0 = δ Z dt ` ( u ; b, D ) = Z dt  δ ` δ u · δ u + δ ` δ b δ b + δ ` δ D δ D  = Z dt  δ ` δ u ·  ∂ w ∂ t − ad u w  − δ ` δ b w · ∇ b − δ ` δ D  ∇ · ( D w )   = Z dt w ·  − ∂ ∂ t δ ` δ u − ad ∗ u δ ` δ u − δ ` δ b ∇ b + D ∇ δ ` δ D  = − Z dt w ·   ∂ ∂ t + £ u  δ ` δ u + δ ` δ b ∇ b − D ∇ δ ` δ D  , (31.5) where we hav e consistently dropp ed b oundary terms arising from integrations by parts, b y inv oking natural b oundary conditions. Sp eciﬁcally , we ma y imp ose ˆ n · w = 0 on the b oundary , where ˆ n is the b oundary’s out w ard unit normal vector and w = δ η t ◦ η − 1 t v anishes at the endp oin ts. 31.2 Euler–P oincar´ e framew ork for GFD The Euler–P oincar´ e equations for con tinua (30.16) ma y now be summarized in v ector form for adv ected Eulerian v ariables a in the set (31.1). W e adopt the notational con ven tion of the circulation map I in equations (30.21) and (30.22) that a one form densit y can b e made into a one form (no longer a densit y) by dividing it by the mass density D and we use the Lie-deriv ativ e relation for the contin uity equation ( ∂ /∂ t + £ u ) D d 3 x = 0. Then, the Euclidean comp onen ts of the Euler–Poincar ´ e equations for con tin ua in equation (31.5) are expressed in Kelvin theorem form (30.23) with a slight abuse of notation as  ∂ ∂ t + £ u  1 D δ ` δ u · d x  + 1 D δ ` δ b ∇ b · d x − ∇  δ ` δ D  · d x = 0 , (31.6) in whic h the v ariational deriv ativ es of the Lagrangian ` are to b e computed according to the usual ph ysical conv entions, i.e., as F r ´ echet deriv ativ es. F ormula (31.6) is the Kelvin–Noether form of the equation of motion for ideal contin ua. Hence, w e hav e the explicit Kelvin theorem expression, cf. equations (30.21) and (30.22), d dt I γ t ( u ) 1 D δ ` δ u · d x = − I γ t ( u ) 1 D δ ` δ b ∇ b · d x , (31.7) where the curve γ t ( u ) mov es with the ﬂuid velocity u . Then, by Stokes’ theorem, the Euler equations generate circulation of v := ( D − 1 δ l/δ u ) whenev er the gradients ∇ b and ∇ ( D − 1 δ l/δ b ) are not collinear. The corresp onding c onservation of p otential vorticity q on ﬂuid parcels is giv en by ∂ q ∂ t + u · ∇ q = 0 , where q = 1 D ∇ b · curl  1 D δ ` δ u  . (31.8) This is also called PV c onve ction . Equations (31.6-31.8) em b o dy most of the panoply of equations for GFD. The v ector form of equation (31.6) is,  ∂ ∂ t + u · ∇  1 D δ l δ u  + 1 D δ l δ u j ∇ u j | {z } Geo desic Nonlinearity: Kinetic energy = ∇ δ l δ D − 1 D δ l δ b ∇ b | {z } P oten tial energy (31.9) D. D. Holm Imp erial College London Applications of P oisson Geometry 126 In geoph ysical applications, the Eulerian v ariable D represents the frozen-in v olume element and b is the buo yancy . In this case, Kelvin’s theorem is dI dt = Z Z S ( t ) ∇  1 D δ l δ b  × ∇ b · d S , with circulation in tegral I = I γ ( t ) 1 D δ l δ u · d x . 31.3 Euler’s Equations for a Rotating Stratiﬁed Ideal Incompressible Fluid The Lagrangian. In the Eulerian velocity represen tation, we consider Hamilton’s principle for ﬂuid motion in a three dimensional domain with action functional S = R l dt and Lagrangian l ( u , b, D ) giv en b y l ( u , b, D ) = Z ρ 0 D (1 + b )  1 2 | u | 2 + u · R ( x ) − g z  − p ( D − 1) d 3 x , (31.10) where ρ tot = ρ 0 D (1 + b ) is the total mass density , ρ 0 is a dimensional constant and R is a given function of x . This v ariations at ﬁxed x and t of this Lagrangian are the following, 1 D δ l δ u = ρ 0 (1 + b )( u + R ) , δ l δ b = ρ 0 D  1 2 | u | 2 + u · R − g z  , δ l δ D = ρ 0 (1 + b )  1 2 | u | 2 + u · R − g z  − p , δ l δ p = − ( D − 1) . (31.11) Hence, from the Euclidean comp onen t formula (31.9) for Hamilton principles of this type and the fundamen tal vector iden tity , ( b · ∇ ) a + a j ∇ b j = − b × ( ∇ × a ) + ∇ ( b · a ) , (31.12) w e ﬁnd the motion equation for an Euler ﬂuid in three dimensions, d u dt − u × curl R + g ˆ z + 1 ρ 0 (1 + b ) ∇ p = 0 , (31.13) where curl R = 2 Ω ( x ) is the Coriolis parameter (i.e., t wice the lo cal angular rotation frequency). In writing this equation, w e hav e used adv ection of buoy ancy , ∂ b ∂ t + u · ∇ b = 0 , from equation (31.2). The pressure p is determined by requiring preserv ation of the constrain t D = 1, for which the contin uity equation (31.3) implies div u = 0. The Euler motion equation (31.13) is Newton’s Law for the acceleration of a ﬂuid due to three forces: Coriolis, gra vity and pressure gradien t. The dynamic balances among these three forces pro duce the man y circulatory ﬂows of geophysical ﬂuid dynamics. The c onservation of p otential vorticity q on ﬂuid parcels for these Euler GFD ﬂo ws is given b y ∂ q ∂ t + u · ∇ q = 0 , where, on using D = 1 , q = ∇ b · curl  u + R  . (31.14) Exercise 31.1 (Semidirect-pro duct Lie-P oisson brac ket for compressible ideal ﬂuids) D. D. Holm Imp erial College London Applications of P oisson Geometry 127 1. Compute the L e gendr e tr ansform for the L agr angian, l ( u , b, D ) : X × Λ 0 × Λ 3 7→ R whose adve cte d variables satisfy the auxiliary e quations, ∂ b ∂ t = − u · ∇ b , ∂ D ∂ t = − ∇ · ( D u ) . 2. Compute the Hamiltonian, assuming the L e gendr e tr ansform is a line ar invertible op er ator on the velo city u . F or deﬁniteness in c omputing the Hamiltonian, assume the L agr angian is given by l ( u , b, D ) = Z D  1 2 | u | 2 + u · R ( x ) − e ( D , b )  d 3 x , (31.15) with pr escrib e d function R ( x ) and sp e ciﬁc internal ener gy e ( D , b ) satisfying the First L aw of Thermo dynamics, de = p D 2 dD + T db , wher e p is pr essur e, T temp er atur e. 3. Find the semidir e ct-pr o duct Lie-Poisson br acket for the Hamiltonian formulation of these e qua- tions. 4. Do es this Lie-Poisson br acket have Casimirs? If so, what ar e the c orr esp onding symmetries and momentum maps? 32 Hamilton-P oincar ´ e reduction and Lie-P oisson equations In the Euler-P oincar´ e framew ork one starts with a Lagrangian deﬁned on the tangent bundle of a Lie group G L : T G → R and the dynamics is giv en by Euler-Lagrange equations arising from the v ariational principle δ Z t 1 t 0 L ( g , ˙ g ) dt = 0 The Lagrangian L is taken left/right inv ariant and because of this prop ert y one can r e duc e the problem obtaining a new system which is deﬁned on the Lie algebra g of G, obtaining a new set of equations, the Euler-P oincar´ e equations, arising from a reduced v ariational principle δ Z t 1 t 0 l ( ξ ) dt = 0 where l ( ξ ) is the reduced lagrangian and ξ ∈ g . Problem 32.1 Is ther e a similar pr o c e dur e for Hamiltonian systems? Mor e pr e cisely: given a Hamil- tonian function deﬁne d on the c otangent bund le T ∗ G H : T ∗ G → R one wants to p erform a similar pr o c e dur e of r e duction and derive the e quations of motion on the dual of the Lie algebr a g ∗ , pr ovide d the Hamiltonian is again left/right invariant. D. D. Holm Imp erial College London Applications of P oisson Geometry 128 Hamilton-P oincar´ e reduction gives a p ositive answer to this problem, in the context of v ariational principles as it is done in the Euler-Poincar ´ e framework: we are going to explain how this pro cedure is p erformed. More in general, we will also consider advected quan tities b elonging to a v ector space V on which G acts, so that the Hamiltonian is written in this case as [ ? ] [HoMaRa98] H : T ∗ G × V ∗ → R The space V is regarded here exactly the same as in the Euler-Poincar ´ e theory . The equations of motion, i.e. Hamilton’s equations, may b e deriv ed from the following v ariational principle δ Z t 1 t 0 {h p ( t ) , ˙ g ( t ) i − H a 0 ( g ( t ) , p ( t )) } dt = 0 as it is w ell know from ordinary classical mechanics ( ˙ g ( t ) has to b e considered as the tangen t vector to the curv e g ( t ), so that ˙ g ( t ) ∈ T g ( t ) G ). Problem 32.2 What happ ens if H a 0 is left/right invariant? It turns out that in this case the whole function F ( g , ˙ g , p ) = h p, ˙ g i − H a 0 ( g , p ) is also inv ariant. The pro of is straightforw ard once the action is sp eciﬁed (from now on we consider only left in v ariance): h ( g , ˙ g , p ) = ( hg , T g L h ˙ g , T ∗ hg L h − 1 p ) where T g L h : T g G → T hg G is the tangen t of the left translation map L h g = hg ∈ G at the point g and T ∗ hg L h − 1 : T ∗ g G → T ∗ hg G is the dual of the map T hg L h − 1 : T hg G → T g G . W e now chec k that h h p, h ˙ g i = h T ∗ hg L h − 1 p, T g L h ˙ g i = h p, T hg L h − 1 ◦ T g L h ˙ g i = h p, T g ( L h − 1 ◦ L h ) ˙ g i = h p, ˙ g i where the c hain rule for the tangent map has been used. The same result holds for the right action. Due to this in v ariance prop ert y , one can write the v ariational principle as δ Z t 1 t 0 {h µ, ξ i − h ( µ, a ) } dt = 0 with µ ( t ) = g − 1 ( t ) p ( t ) ∈ g ∗ , ξ ( t ) = g − 1 ( t ) ˙ g ( t ) ∈ g , a ( t ) = g − 1 ( t ) a 0 ∈ V ∗ In particular a ( t ) is the solution of ˙ a ( t ) = − ξ ( t ) a 0 . where a Lie algebra action of g on V ∗ is implicitly deﬁned. In order to ﬁnd the equations of motion one calculates the v ariations δ Z t 1 t 0 {h µ, ξ i − h ( µ, a ) } dt = Z t 1 t 0  h δ µ, ξ i + h µ, δ ξ i −  δ µ, δ h δ µ  −  δ a, δ h δ a  dt D. D. Holm Imp erial College London Applications of P oisson Geometry 129 As in the Euler-P oincar´ e theorem, we use the follo wing expressions for the v ariations δ ξ = ˙ η + [ ξ , η ] , δa = − η a and using the deﬁnition of the diamond op erator w e ﬁnd Z t 1 t 0  h δ µ, ξ i + h µ, δ ξ i −  δ µ, δ h δ µ  −  δ a, δ h δ a  dt = Z t 1 t 0  δ µ, ξ − δ h δ µ  + h µ, ˙ η + ad ξ η i +  η a, δ h δ a  dt = Z t 1 t 0  δ µ, ξ − δ h δ µ  + h− ˙ µ + ad ∗ ξ µ, η i −  δ h δ a  a, η  dt so that ξ = δ h δ µ and the equations of motion are ˙ µ = ad ∗ ξ µ − δ h δ a  a together with ˙ a = − δ h δ µ a. This equations of motion written on the dual Lie algebra g are called Lie-Poisson equations. W e hav e no w prov en the follo wing Theorem 32.3 [Hamilton-Poinc ar´ e r e duction the or em] With the pr e c e ding notation, the fol lowing statements ar e e quivalent: 1. With a 0 held ﬁxe d, the variational principle δ Z t 1 t 0 {h p ( t ) , ˙ g ( t ) i − H a 0 ( g ( t ) , p ( t )) } dt = 0 holds, for variations δ g ( t ) of g ( t ) vanishing at the endp oints. 2. ( g ( t ) , p ( t )) satisﬁes Hamilton ’s e quations for H a 0 on G. 3. The c onstr aine d variational principle δ Z t 1 t 0 {h µ ( t ) , ξ ( t ) i − h ( µ ( t ) , a ( t )) } dt = 0 holds for g × V ∗ , using variations of ξ and a of the form δ ξ = ˙ η + [ ξ , η ] , δ a = − η a wher e η ( t ) ∈ g vanishes at the endp oints D. D. Holm Imp erial College London Applications of P oisson Geometry 130 4. The Lie-P oisson e quations hold on g × V ∗ ( ˙ µ, ˙ a ) =  ad ∗ ξ µ − δ h δ a  a, − δ h δ µ a  Remark 32.4 Mor e exactly one should start with an invariant Hamiltonian deﬁne d on T ∗ ( G × V ) = T ∗ G × V × V ∗ However, as mentione d in [][HoMaR e98], such an appr o ach turns out to b e e quivalent to the tr e atment pr esente d her e. Remark 32.5 [Legendre transform] Lie-Poisson e quations may arise fr om the Euler-Poinc ar´ e setting by L e gendr e tr ansform µ = δ l δ ξ . If this is a diﬀe omorphism, then the Hamilton-Poinc ar´ e the or em is e quivalent to the Euler-Poinc ar´ e the or em. Remark 32.6 [Lie-Poisson structure] One shows that g ∗ × V ∗ is a Poisson manifold: ˙ F ( µ, a ) =  ˙ µ, δ F δ µ  +  ˙ a, δ F δ a  = =  ad ∗ δ H/δ µ µ − δ H δ a  a, δ F δ µ  −  δ H δ µ a, δ F δ a  = =  µ,  δ H δ µ , δ F δ µ  −  δ H δ a  a, δ F δ µ  −  δ H δ µ a, δ F δ a  = = −  µ,  δ F δ µ , δ H δ µ  −  δ H δ a  a,  −  δ H δ µ a, δ F δ a  = = −  µ,  δ F δ µ , δ H δ µ  −  a, δ F δ µ δ H δ a − δ H δ µ δ F δ a  In fact it c an b e e asily shown that this structur e { F , H } ( µ, a ) = −  µ,  δ F δ µ , δ H δ µ  −  a, δ F δ µ δ H δ a − δ H δ µ δ F δ a  satisﬁes the deﬁnition of a Poisson structur e. In p articular one ﬁnds that any dual Lie algebr a g is a Poisson manifold . Ple ase note : this structur e has b e en found during le ctur es for the simpler c ase without adve cte d quantities. Remark 32.7 [right inv ariance] It c an b e shown that for a right invariant Hamiltonian one has { F , H } ( µ, a ) = +  µ,  δ F δ µ , δ H δ µ  +  a, δ F δ µ δ H δ a − δ H δ µ δ F δ a  ( ˙ µ, ˙ a ) = −  ad ∗ ξ µ − δ h δ a  a, − δ h δ µ a  with al l signs change d r esp e ct to the c ase of left invarianc e. D. D. Holm Imp erial College London Applications of P oisson Geometry 131 33 Tw o applications 33.1 The Vlasov equation In plasma ph ysics a main topic is collisionless particle dynamics, whose main equation, the Vlasov equation, will be heuristically derived here. In this con text a cen tral role is held b y the distribution function on phase space f ( q , p , t ), basically expressing the particle density on phase space. Intended as a density one deﬁnes F := f ( q , p , t ) d q d p : b ecause of the conserv ation of particles, one writes the con tinuit y equation just as one do es as in the con text of ﬂuid dynamics ˙ F + ∇ · ( u F ) = 0 where u is a “velocity” v ector ﬁeld on phase space, whic h is given b y the single particle motion u = ( ˙ q , ˙ p ) ∈ X ( T ∗ R N ) if we now assume that the generic single particle undergo es a Hamiltonian motion, the Hamiltonian function h ( q , p , t ) can be in tro duced directly b y means of the single particle Hamilton’s equations ( ˙ q , ˙ p ) =  ∂ h ∂ p , − ∂ h ∂ q  whic h shows that u has zero divergence, assuming the Hessian of h is symmetric. Therefore, the Vlaso v equation written in terms of the distribution function f ( q , p , t ) is ˙ f + u · ∇ f = 0 Expanding no w the Hamiltonian h as the total single particle energy h ( q , p , t ) = 1 2 m p 2 + V ( q , p , t ) one obtains the more common form ∂ f ∂ t + p m · ∂ f ∂ q − ∂ V ∂ q · ∂ f ∂ p = 0 Problem 33.1 Can Vlasov e quation b e c ast in Lie-Poisson form? W e show here why the answer is yes. First we write the Vlasov equation in terms of a generic single particle Hamiltonian h as ˙ f + { f , h } = 0 where w e recall the canonical Poisson brac k et { f , h } = ∂ f ∂ q · ∂ h ∂ p − ∂ f ∂ p · ∂ h ∂ q The main p oin t of this discussion is that the canonical Poisson brack et provides the set F ( T ∗ R N ) of the functions on the phase space with a Lie algebra structure [ k , h ] = { k , h } D. D. Holm Imp erial College London Applications of P oisson Geometry 132 A t this p oin t, in order to lo ok for a Lie-P oisson equation, one calculates the coadjoint op erator such that h f , { h, k }i = h f , ad h k i = h ad ∗ h f , k i = h−{ h, f } , k i where the last equalit y is justiﬁed b y the Leibniz property of the P oisson brack et, with the pairing deﬁned as h f , g i = Z f g d q d p . In conclusion, the argumen t abov e sho ws that the Vlasov equation can in fact be written in the Lie P oisson form ˙ f + ad ∗ h f = 0 33.2 Ideal barotropic compressible ﬂuids The reduced Lagrangian for ideal compressible ﬂuids is written as l ( u , D ) = Z D 2 | u | 2 − De ( D ) d x where u ∈ X ( M ⊂ R 3 ) is tangential on the b oundary ∂ M and D is the advected densit y , which satisﬁes the c ontinuity e quation ∂ t D + L u D = 0 . Moreo ver, the in ternal e nergy satisﬁes the barotropic First Law of Thermodynamics de = − p ( D ) d ( D − 1 ) = p ( D ) D 2 dD for the pressure p ( D ). The “reduced” Legendre transform on this Lie algebra X ( R 3 ) is giv en by m = D u and the Hamiltonian is then written as h ( m , D ) = h m , u i − l ( u , D ) that is h ( m , D ) = Z 1 2 D | m | 2 + De ( D ) d x The Lie P oisson equations in this case are as from the general theory ∂ t m = − ad ∗ δ h/δ m m − δ h δ D  D ∂ t D = −L δ h/δ m D Earlier we found that the coadjoint action is given b y the Lie deriv ative. On the other hand we ma y calculate the expression of the diamond op eration from its deﬁnition  δ h δ D , − £ η D  =  δ h δ D  D , η  D. D. Holm Imp erial College London Applications of P oisson Geometry 133 to b e  δ h δ D , − div D η  =  D ∇ δ h δ D , η  Therefore, w e hav e δ h δ D  D = D ∇ δ h δ D where δ h/δ D = − | m | 2 2 D 2 +  e + p D  Substituting into the momen tum equation and using the First Law to ﬁnd d ( e + p/D ) = (1 /D ) dp yields ∂ t m = −L u m − ∇ p Up on expanding the Lie deriv ative for the momentum densit y m and using the contin uity equation for the densit y , this quickly becomes ∂ t u = − u · ∇ u − 1 D ∇ p whic h is Euler’s equation for a barotropic ﬂuid. 33.3 Euler’s equations for ideal incompressible ﬂuid motion The barotropic equations recov er Euler’s equations for ideal incompressible ﬂuid motion when the in ternal energy in the reduced Lagrangian for ideal compressible ﬂuids is replaced b y the constrain t D = 1, as l ( u , D ) = Z D 2 | u | 2 − p ( D − 1) d x where again u ∈ X ( M ⊂ R 3 ) is tangential on the boundary ∂ M and the advected densit y D satisﬁes the con tinuit y equation, ∂ t D + div D u = 0 . This equation enforces incompressibility div u = 0 when ev aluated on the constraint D = 1. The pressure p is now a Lagrange multiplier, whic h is determined b y the condition that incompressibility b e preserv ed b y the dynamics. References [AbMa1978] Abraham, R. and Marsden, J.E. [1978], F oundations of Me chanics . Addison-W esley , second edition. [AcHoKoTi1997] Acev es, A., D. D. Holm, G. Ko v acic and I. Timofeyev [1997] Homo clinic Orbits and Chaos in a Second-Harmonic Generating Optical Ca vity . Phys. L ett. A 233 203-208. [AbMaRa1988] Abraham, R., Marsden, J.E. and Ratiu, T.S. [1988], Manifolds, T ensor A nalysis, and Applic ations . V olume 75 of Applie d Mathematic al Scienc es . Springer-V erlag, second edition. [AlHo1996] Allen, J. S., and Holm, D. D. [1996] Extended-geostrophic Hamiltonian mo dels for rotating shallo w water motion. Physic a D , 98 229-248. D. D. Holm Imp erial College London Applications of P oisson Geometry 134 [AlHoNe2002] Allen, J. S., Holm, D. D., and Newb erger, P . A. [2002] T o ward an extended-geostrophic Euler–P oincar´ e mo del for mesoscale o ceanographic ﬂo w. In L ar ge-Sc ale Atmospher e-Oc e an Dy- namics 1: Analytic al Metho ds and Numeric al Mo dels . Edited by J. Norbury & I. Roulstone, Cam bridge Universit y Press: Cambridge, pp. 101–125. [AmCZ96] Am brosetti, A. and Coti Zelati, V. [1996], Perio dic Solutions of Singular L agr angian Sys- tems . Birkh¨ auser [AnMc1978] Andrews, D. G. and McInt yre, M. E. [1978] An exact theory of nonlinear w av es on a Lagrangian-mean ﬂo w. J. Fluid Me ch. 89 , 609–646. [Ar1966] Arnold V.I. [1966], Sur la g´ eometrie diﬀerentialle des group es de Lie de dimiension inﬁnie et ses applications ` a l’hydrodynamique des ﬂuids parfaits. Ann. Inst. F ourier, Gr enoble 16 , 319-361. [Ar1979] Arnold V.I. [1979], Mathematic al Metho ds of Classic al Me chanics . V olume 60 of Gr aduate T exts in Mathematics . Springer-V erlag. [ArKh1998] Arnold, V. I and Khesin, B. A. [1998] T op olo gic al Metho ds in Hydr o dynamics. Springer: New Y ork. [Be1986] Benci, V. [1986], P erio dic solutions of Lagrangian Systems on a compact manifold, Journal of Diﬀer ential Equations 63 , 135–161. [BlBr92] Blanc hard, P . and B runing, E. [1992], V ariational Metho ds in Mathematic al Physics . Springer V erlag. [Bl2004] Blo c h, A. M. (with the collab oration of: J. Baillieul, P . E. Crouch, and J. E. Marsden) [2004] Nonholonomic Me chanics and Contr ol , Springer-V erlag NY. [BlBrCr1997] Blo c h, A.M., R. W. Bro c kett, and P .E. Crouch [1997] Double brac ket equations and geo desic ﬂo ws on symmetric spaces. Comm. Math Phys 187 ,357-373. [BlCr1997] Blo c h, A. M. and P . E. Crouc h [1996] Optimal control and geo desic ﬂo ws. Systems and Contr ol L etters 28 , 65-72. [BlCrHoMa2001] Blo c h, A.M., P .E. Crouch, D. D. Holm and J. E. Marsden [2001] Pro c. of the 39th IEEEE Conference on Decision and Con trol, Pr o c. CDC 39 , 12731279. [Bo71] Bourbaki, N. [1971], V ari´ et´ es diﬀ ´ er entiel les et analytiques. F ascicule de r´ esultats . Hermann. [Bo1989] Lie 1-3 Bourbaki, N. [1989], Lie Gr oups and Lie Algebr as. Chapters 1–3. Springer V erlag. [Ca1995] Calogero, F. [1995] An in tegrable hamiltonian system. Phys. L ett. A 201 , 306-310. [CaF r1996] Calogero, F. and F rancoise, J.-P . [1996] An integrable hamiltonian system. J. Math. Phys. 37 , 2863-2871. [CaHo1993] Camassa, R. and Holm, D. D. [1993] An integrable shallow water equation with p eak ed solitons. Phys. R ev. L ett. 71 , 1661-64. [CeMaP eRa2003] Cendra, H, Marsden, J.E., P ek arsky , S., and Ratiu, T.S. [2003], V ariational princi- ples for Lie-P oisson and Hamilton-Poincar ´ e equations, Moskow Math. Journ. , 3 (3), 833–867. [Ch1969] Chandrasekhar, S. [1969] El lipsoidal Figur es of Equilibrium . Y ale Universit y Press. D. D. Holm Imp erial College London Applications of P oisson Geometry 135 [ChMa1974] Chernoﬀ, P . R. and Marsden, J.E. [1974], Pr op erties of Inﬁnite Dimensional Hamiltonian Systems , Lecture Notes in Mathematics, 425 , Springer-V erlag. [CoLe1984] Co ddington, E. A. and Levinson, N. The ory of Or dinary Diﬀer ential Equations , Mel- b ourne, FL: Robert E. Krieger, 1984. [CuBa1997] Cushman, R.H. and Bates, L.M. [1997], Glob al Asp e cts of Inte gr able Systems . Birkh¨ auser. [DaHo1992] Da vid, D. and D. D. Holm, [1992] Multiple Lie-Poi sson Structures, Reductions, and Geometric Phases for the Maxw ell-Blo c h T rav eling-W a ve Equations. J. Nonlin. Sci. 2 241–262. [De1860] Dedekind, R. [1860] Zusatz zu der v orstehenden Abhandlung. J. Riene A ngew. Math. 58 , 217-228. [DeMe1993] Dellnitz M. and Melb ourne I. [1993], The equiv ariant Darb oux Theorem, L e ctur es in Appl. Math. , 29 , 163–169. [DrJo1989] Drazin, P .G. and Johnson, R.S. [1989], Solitons: An Intr o duction . Cambridge Univ ersity Press. [DuNoF o1995] Dubrovin, B., No viko v, S.P ., and F omenko, A.T. [1995], Mo dern Ge ometry I, II, III , V olumes 93, 104, 124 of Gr aduate T exts in Mathematics . Springer-V erlag. [DuGoHo2001] Dullin, H., Gottw ald, G. and Holm, D. D. [2001] An integrable shallo w water equation with linear and nonlinear disp ersion. Phys. R ev. L ett. 87 194501-04. [DuGoHo2003] Dullin, H., Gottw ald, G. and Holm, D. D. [2003] Camassa-Holm, Korteweg-de V ries- 5 and other asymptotically equiv alen t equations for shallo w w ater wa ves. Fluid Dyn. R es. 33 (2003) 7395. [DuGoHo2004] Dullin, H., Gottw ald, G. and Holm, D. D. [2003] On asymptotically equiv alen t shallow w ater wa ve equations. Physic a D 190 (2004) 1-14. [EbMa1970] Ebin, D.G. and Marsden, J.E. [1970], Groups of diﬀeomorphisms and the motion of an incompressible ﬂuid, A nn. of Math. 92 , 102–163. [Er1942] Ertel, H. [1942] Ein Neuer Hydro dynamisc her Wirb elsatz, Met. Z., 59 , 271–281. [F oHo1991] F ordy , A.P . and Holm, D. D. [1991] A T ri-Hamiltonian F orm ulation of the Self-Induced T ransparency Equations. Phys. L ett A 160 143–148. [F rHo2001] F ringer, O. B. and Holm, D. D. [2001] Integrable vs nonintegrable geo desic soliton behav- ior. Physic a D 150 237-263. [F u1996] F uchssteiner, B. [1996] Some tricks from the symmetry-to olb o x for nonlinear equations: generalization of the CamassaHolm equation. Physic a D 95 , 229-243. [GeDo1979] Gelfand, I. M. and Dorfman, I. Y a. R. [1979] F unct. Anal. Appl. 13 , 248. [GiHoKu1982] Gibb ons, J., Holm, D. D. and B. A. Kup ershmidt [1982], Gauge-Inv ariant P oisson Brac kets for Chromoh ydro dynamics, Phys. L ett. A 90 281–283. [GjHo1996] Gja ja, I. and Holm, D. D. Self-consistent w av e-mean ﬂow in teraction dynamics and its Hamiltonian formulation for a rotating stratiﬁed incompressible ﬂuid. Physic a D , 98 (1996) 343-378. D. D. Holm Imp erial College London Applications of P oisson Geometry 136 [GuSt1984] Guillemin, V. and Sternberg, S. [1984], Symple ctic T e chniques in Physics . Cambridge Univ ersity Press. [Ha1984] Haine, L. [1984] The algebraic complete integrabilit y of geodesic ﬂo w on SO(4). Commun, Math, Phys. 94 , 271-287. [Ho1983] Holm, D.D. [1983] Magnetic tornado es: three-dimensional aﬃne motions in ideal magneto- h ydro dynamics, Physic a D 8 , 170-182. [Ho1991] Holm, D. D. [1991] Elliptical v ortices and integrable Hamiltonian dynamics of the rotating shallo w-water equations. J. Fluid Me ch. , 227 393-406. [Ho1996a] Holm, D. D. [1996a] Hamiltonian balance equations. Physic a D , 98 379-414. [Ho1996b] Holm, D. D. [1996b] The ideal Craik-Leib o vic h equations. Physic a D , 98 (1996) 415-441. [Ho1999] Holm, D. D. [1999] Fluctuation eﬀects on 3D Lagrangian mean and Eulerian mean ﬂuid motion. Physic a D 133 , 215–269. [Ho2002] Holm, D. D. [2002] Averaged Lagrangians and the mean dynamical eﬀects of ﬂuctuations in con tinuum mec hanics. Physic a D 170 , 253–286. [Ho2002a] Holm, D. D. [2002] Euler-Poincar ´ e dynamics of p erfect complex ﬂuids. In Ge ometry, Me- chanics, and Dynamics: in honor of the 60th birthday of Jerr old E. Marsden edited by P . Newton, P . Holmes and A. W einstein. Springer, pp. 113-167. [Ho2002b] Holm, D. D. [2002] V ariational principles for Lagrangian av eraged ﬂuid dynamics. J. Phys. A: Math. Gen. 35 1-10. [Ho2002c] Holm, D. D. [2002] Lagrangian a verages, a veraged Lagrangians, and the mean eﬀects of ﬂuctuations in ﬂuid dynamics. Chaos 12 518-530. [Ho2005] Holm, D. D. [2005] The Euler-Poincar ´ e v ariational framework for modeling ﬂuid dynamics, in Ge ometric Me chanics and Symmetry: The Peyr esq L e ctur es , edited by J. Montaldi and T. Ratiu, London Mathematical So ciet y Lecture Notes Series 306, Cam bridge Universit y Press (2005). [HoKo1991] Holm, D. D. and G. Ko v acic [1991] Homo clinic Chaos for Ra y Optics in a Fib er. Physic a D 51 , 177–188. [HoKo1992] Holm, D. D. and G. Ko v acic [1992] Homo clinic Chaos in a Laser-Matter System . Physic a D 56 270–300. [HoKu1982] Holm, D. D. and B. A. Kup ershmidt [1982], Poisson Structures of Superﬂuids, Phys. L ett. A 91 425–430. [HoKu1983] Holm, D. D. and B. A. Kup ershmidt [1983], Poisson brack ets and Clebsch representations for magnetoh ydro dynamics, m ultiﬂuid plasmas, and elasticity , Physic a D 6 , 347–363. [HoKu1988] Holm, D. D. and B. A. Kup ershmidt [1988], The Analogy Betw een Spin Glasses and Y ang-Mills Fluids, J. Math Phys. 29 21–30. D. D. Holm Imp erial College London Applications of P oisson Geometry 137 [HoMa2004] Holm, D.D. and J. E. Marsden [2004] Momentum maps and measure v alued solutions (p eak ons, ﬁlaments, and sheets) of the Euler-Poincare equations for the diﬀeomorphism group. In The Br e adth of Symple ctic and Poisson Ge ometry , (Marsden, J. E. and T. S. Ratiu, eds) Birkh¨ auser Boston, to app ear (2004). h [HoMaRa1986] Holm, D. D., Marsden, J. E. and Ratiu, T. S. [1986] The Hamiltonian Structure of Con tinuum Mec hanics in Material, Inv erse Material, Spatial and Conv ective Represen tations. In Hamiltonian Structur e and Lyapunov Stability for Ide al Continuum Dynamics , Univ. Montreal Press, pp. 1-124. [HoMaRa1998a] Holm, D. D., Marsden, J. E. and Ratiu, T. S. [1998a] The Euler–P oincar´ e equations and semidirect pro ducts with applications to con tin uum theories. A dv. in Math. 137 , 1–81. [HoMaRa1998b] Holm, D. D., Marsden, J. E. and Ratiu, T. S. [1998b] Euler–Poincar ´ e mo dels of ideal ﬂuids with nonlinear disp ersion. Phys. R ev. L ett. 80 , 4173–4177. [HoMaRa2002] Holm, D. D., Marsden, J. E. and Ratiu, T. S. [2002] The Euler–Poincar ´ e equations in geoph ysical ﬂuid dynamics. In L ar ge-Sc ale Atmospher e-Oc e an Dynamics 2: Ge ometric Metho ds and Mo dels . Edited b y J. Norbury and I. Roulstone, Cam bridge Univ ersit y Press: Cam bridge, pp. 251–299. [HoMaRaW e1985] Holm, D. D., Marsden, J. E., Ratiu, T. S. and W einstein, A. [1985] Nonlinear stabilit y of ﬂuid and plasma equilibria. Physics R ep orts 123 1-116. [HoRaT rY o2004] Holm, D. D., J. T. Rananather, A. T rouv´ e and L. Y ounes [2004] Soliton Dynamics in Computational Anatom y . Neur oImage 23 , S170-178. [HoSt2002] Holm, D. D. and Staley , M. F. [2002] W av e Structures and Nonlinear Balances in a F amily of 1+1 Ev olutionary PDEs. http://arxiv.org/abs/nlin.CD/0202059. Submitted to SIADS. [HoSt2004] Holm, D. D. and Staley , M. F. [2004] Interaction Dynamics of Singular W av e F ronts. Submitted to SIAM J. Appl. Dyn. Syst. [HoW o1991] Holm, D. D. and K.B. W olf [1991] Lie-Poisson Description of Hamiltonian Ra y Optics. Physic a D 51 , 189–199. [HuZh1994] Hun ter, J. K. and Zheng, Y. [1994] On a completely integrable nonlinear hyperb olic v ariational equation. Physic a D 79 , 361-386. [JoSa98] Jos ´ e, J.V. and Saletan, E.J. [1998], Classic al Dynamics : A Contemp or ary Appr o ach . Cam- bridge Univ ersity Press. [Jo1998] Jost, J. [1998], R iemannian Ge ometry and Ge ometric Analysis . University T ext . Springer- V erlag, second edition. [KaKoSt1978] Kazhdan, D. Kostant, B., and Sternberg, S. [1978], Hamiltonian group actions and dynamical systems of Calogero t yp e, Comm. Pur e Appl. Math. , 31 , 481–508. [KhMis03] Khesin, B. and Misiolek, G. [2003], Euler equations on homogeneous spaces and Virasoro orbits, A dv. in Math. , 176 , 116–144. [Ko1966] Kostan t, B. [1966] Orbits, symplectic structures and represen tation theory . Pr o c. US-Jap an Seminar on Diﬀ. Ge om., Kyoto. Nipp on Hyr onsha, T okyo 77 . D. D. Holm Imp erial College London Applications of P oisson Geometry 138 [KrSc2001] Kruse, H. P ., Sc hreule, J. and Du, W. [2001] A tw o-dimensional version of the CH equation. In Symmetry and Perturb ation The ory: SPT 2001 Edited b y D. Bam busi, G. Gaeta and M. Cadoni. W orld Scientiﬁc: New Y ork, pp 120-127. [La1999] Lang, S. [1999], F undamentals of Diﬀer ential Ge ometry . V olume 191 of Gr aduate T exts in Mathematics . Springer-V erlag, New Y ork. [Le2003] Lee, J. [2003] Intr o duction to Smo oth Manifolds , Springer-V erlag. [LiMa1987] Lib ermann, P . and Marle, C.-M. [1987], Symple ctic Ge ometry and Analytic al Me chanics . Reidel. [Lie1890] Lie, S. Theorie der T ransformationsgrupp en. Zw eiter Abschnitt. T eubner. [Man1976] Manak ov, S.V. [1976] Note on the in tegration of Euler’s equations of the dynamics of and n -dimensional rigid b ody . F unct. Anal. and its Appl. 10 , 328–329. [Mar1976] Marle, C.-M. [1976], Symplectic manifolds, dynamical groups, and Hamiltonian mechanics, in Diﬀer ential Ge ometry and R elativity , Cahen, M. and Flato, M. eds., D. Reidel, Boston, 249– 269. [Ma1981] Marsden, J.E. [1981], L e ctur es on Ge ometric Metho ds in Mathematic al Physics . V olume 37, SIAM, Philadelphia, 1981. [Ma1992] Marsden, J.E. [1992], L e ctur es on Me chanics . V olume 174 of L ondon Mathematic al So ciety L e ctur e Note Series . Cambridge Universit y Press. [MaHu1983] Marsden, J. E. and T. J. R. Hughes [1983], Mathematic al F oundations of Elasticity . Pren tice Hall. Reprinted by Do v er Publications, NY, 1994. [MaMiOrP eRa2004] Marsden, J.E., Misiolek, G., Ortega, J.-P ., Perlm utter, M., and Ratiu, T.S. [2004], Hamiltonian R e duction by Stages . Lecture Notes in Mathematics, to app ear. Springer- V erlag. [MaMoRa90] Marsden, J.E., Mon tgomery , R., and Ratiu, T.S. [1984], Reduction, symmetry , and phases in mec hanics, Memoirs Amer. Math. So c. , 88 (436), 1–110. [MaRa1994] Marsden, J.E. and Ratiu, T.S. [1994], Intr o duction to Me chanics and Symmetry . V ol- ume 75 of T exts in Applie d Mathematics , second printing of second edition 2003. Springer-V erlag. [MaRa95] Marsden, J.E. and Ratiu, T.S. [2003], Ge ometric Fluid Dynamics . Unpublished notes. [MaRa03] Marsden, J.E. and Ratiu, T.S. [2003], Me chanics and Symmetry. R e duction The ory . In preparation. [MaRaW e84a] Marsden, J.E., Ratiu, T.S., and W einstein, A. [1984a], Semidirect pro ducts and reduc- tion in mec hanics, T r ans. Amer. Math. So c. , 281 (1), 147–177. [MaRaW e84b] Marsden, J.E., Ratiu, T.S., and W einstein, A. [1984b], Reduction and Hamiltonian structures on duals of semidirect pro duct Lie algebras, Contemp or ary Math. , 28 , 55–100. [MaW e74] Marsden, J.E. and W einstein, A. [1983], Reduction of symplectic manifolds with symmetry , R ep. Math. Phys. , 5 , 121–130. D. D. Holm Imp erial College London Applications of P oisson Geometry 139 [MaW e1983] Marsden, J.E. and W einstein, A. [1983], Coadjoint orbits, vortices and Clebsch v ariables for incompressible ﬂuids, Physic a D , 7 , 305–323. [MaWil989] Ma whin, J. and Willem, M. [1989], Critic al Point The ory and Hamiltonin Systems . V ol- ume 74 of Applie d Mathematic al Scienc es . Springer-V erlag, second edition. [McSa1995] McDuﬀ, D. and Salamon, D. [1995], Intr o duction to Symple ctic T op olo gy . Clarendon Press. [MeDe1993] Melb ourne, I. and Dellnitz, M. [1993], Normal forms for linear Hamiltonian vector ﬁelds comm uting with the action of a compact Lie group, Pr o c. Camb. Phil. So c. , 114 , 235–268. [Mi1963] Milnor, J. [1963], Morse The ory . Princeton Universit y Press. [MiF o1978] Mishchenk o, A. S. and F omenk o, A. T.: Euler equations on ﬁnitedimensional Lie groups, Izv. Acad. Nauk SSSR, Ser. Matem. 42, No.2, 396-415 (1978) (Russian); English translation: Math. USSR-Izv. 12, No.2, 371-389 (1978). [Mi2002] Misiolek, G. [2002], Classical solutions of the p eriodic Camassa-Holm equation, Ge om. F unct. A nal. , 12 , 1080–1104. [No1918] No ether, E. [1918] Nachrichten Gesel l. Wissenschaft. G¨ ottingen 2 235. See also C. H. Kim- b erling [1972] A m. Math. Monthly 79 136. [Ohk1993] Ohkitani, K. [1993] Eigenv alue problems in three-dimensional Euler ﬂows. Phys. Fluids A, 5 , 2570–2572. [Ol2000] Olv er, P . J. [2000] Applic ations of Lie Gr oups to Diﬀer ential Equations . Springer: New Y ork. [OrRa2004] Ortega, J.-P . and Ratiu, T.S. [2004], Momentum Maps and Hamiltonian R e duction . V ol- ume 222 of Pr o gr ess in Mathematics . Birkh¨ auser. [OKh87] Ovsienk o, V.Y. and Khesin, B.A. [1987], Kortew eg-de V ries sup erequations as an Euler equation, F unct. Anal. Appl. , 21 , 329–331. [P a1968] Palais, R. [1968], F oundations of Glob al Non-Line ar A nalysis , W. A. Benjamin, Inc., New Y ork-Amsterdam. [P o1901] Poincar ´ e, H. [1901] Sur une forme nouvelle des ´ equations de la m´ echanique C.R. A c ad. Sci. 132 , 369–371. [Ra1980] Ratiu, T. [1980] The motion of the free n-dimensional rigid b ody . Indiana U. Math. J. 29 , 609-627. [RaT uSbSoT e2005] Ratiu, T.S., R. T udoran, L. Sbano, E. Sousa Dias, G. T erra, A Crash course in Geometric Mechanics, in Ge ometric Me chanics and Symmetry: The Peyr esq L e ctur es , edited by J. Montaldi and T. Ratiu, London Mathematical So ciet y Lecture Notes Series 306, Cambridge Univ ersity Press (2005). [Ri1860] Riemann, B. [1860] Untersuc hungen ¨ ub er die Bewegungen eines ﬂ ¨ ussigen gleic hartigen Ellip- soides. A bh. d. K¨ onigl. Gesel l. der Wis. zu G¨ ottingen 9 , 3-36. [Sc1987] Sc hmid, R. [1987], Inﬁnite Dimensional Hamiltonian Systems . Bibliop olis. D. D. Holm Imp erial College London Applications of P oisson Geometry 140 [Se1992] Serre, J.-P . [1992], Lie A lgebr as and Lie Gr oups , V olume 1500 of L e ctur e Notes in Mathe- matics . Springer-V erlag. [Sh1998] Shk oller, S. 1998 Geometry and curv ature of diﬀeomorphism groups with H 1 metric and mean h ydro dynamics J. F unct. Anal., 160 , 337-365. [Sh2000] Shk oller, S. 2000 Analysis on groups of diﬀeomorphisms of manifolds with b oundary and the a veraged motion of a ﬂuid J. Diﬀer ential Ge ometry, 55 , 145-191. [Sm1970] Smale, S. T op ology and Mec hanics, Inv. Math. 10 , 305-331; 11 , 45-64. [So1970] Souriau, J. M. [1970] Structur e des Syst` emes Dynamiques , Duno d, P aris. [Sp1979] Spiv ak, M. [1979], Diﬀer ential Ge ometry , V olume I. New printing with corrections. Publish or P erish, Inc. Houston, T exas. [V a1996] V aisman, I. [1996], L e ctur es on the Ge ometry of Poisson Manifolds . V olume 118 of Pr o gr ess in Mathematics , Birkh¨ auser. [W a1983] W arner, F.W. [1983], F oundation of Diﬀer entiable Manifolds and Lie Gr oups . V olume 94 of Gr aduate T exts in Mathematics , Springer-V erlag. [W e1983] W einstein, A. [1983a], Sophus Lie and symplectic geometry , Exp osition Math. 1 , 95-96. [W e1983b] W einstein, A. [1983b], The lo cal structure of P oisson manifolds, Journ. Diﬀ. Ge om 18 , 523–557. [W e2002] W einstein, A. [2002] Geometry of momentum (preprin t); ArXiv:math/SG0208108 v1.

Applications of Poisson Geometry to Physical Problems

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment