On Lagrangian and Hamiltonian systems with homogeneous trajectories

On Lagrangian and Hamiltonian systems with homogeneous tra jectories G´ ab or Zsolt T´ oth KFKI R ese ar ch Institute for Particle and Nucle ar P hysics, Hungarian A c ademy of Scienc es, P.O .B. 49, 1525 Bu dap est, Hungary email: tgzs@rmk i.kfki.hu, tgzs@cs.elt e.hu Abstract Motiv ated b y v arious results on homogeneous geo desics of Riemannian spaces, w e study homoge- neous tra jectories, i.e. tra jecto ries which are orbits of a one-parameter s ymmetry group, of La- grangia n and Hamiltonia n s ystems. W e present cr iteria under which an orbit of a one-par ameter subgroup of a symmetry g roup G is a solution of the Euler-La grange or Hamiltonian equations. In particular , w e generalize the ‘g eodesic lemma’ kno wn in Riemannian geo metry to La grangian and Hamiltonian systems. W e present r e s ults on the existence of homogeneous tra jectories o f Lagra ng ian systems. W e study Hamiltonia n a nd La grangian g.o . space s , i.e . homogeneous spaces G/H with G -inv aria n t Lagrang ian or Hamiltonian functions on w hich every s olution o f the equa- tions of motion is homogeneous. W e show that the Hamiltonian g.o. spaces are related to the functions that a re inv aria nt under the c oadjoint a c tio n of G . Riemannian g .o. spaces thus corre- sp ond to sp e cial Ad ∗ ( G )-in v ariant functions. An Ad ∗ ( G )-in v ariant function that is related to a g.o . space also serves as a p otential for the mapping called ‘g eodesic gra ph’. As illustra tio n w e discus s the Riemannia n g.o. metr ic s on S U (3) /S U (2). Keywor ds : g.o. space, homogeneous sp ace, relativ e equilibrium , momentum map, La- grangian and Hamiltonian systems w ith symmetry P A CS n umbers: 45.20. Jj, 02.40.Ky , 02.40. Ma 1 In tro duc tion Let M b e a Riemannian manifold. A geo desic in M is call ed homo gene ous if it is the orbit of a one-parameter group of isometries of M . A homogeneous Riemannian m anifold M = G/K , w here G is a connected Lie group and K is a closed sub group, is a ge o desic orbit (g.o.) sp ac e with resp ect to G , if ev ery geo desic in it is the orb it of a one-parameter subgroup of G . The homogeneous space M = G/K is called a r e ductive sp ac e , if there exists a direct sum decomposition (called r e ductive de c omp osition ) g = m ⊕ k of the Lie algebra of G , where m is an ad ( K )-inv ariant linea r subspace of g and k is the Lie a lgebra of K . It is kn o wn that all R iemann ian homogeneous spaces are reductiv e. If M = G/ K is Riemannian an d there exists a reductiv e d ecomposition g = m ⊕ k such that eac h geo desic in M starting at the origin o ∈ M is an orbit of a one-parameter subgroup of G generated by some elemen t of m , then M is called a natur al ly r e ductive sp ac e w ith resp ect to G , and m is called a natur al c omplement . The origin o is the image of K b y the canonical pro jection G → G/K . Ob viously , ev ery naturally reductive space is a g.o. sp ace as w ell. It w as b eliev ed some decades ago that the con verse is also true, i.e. ev ery g.o. space is isometric to some naturally red u ctiv e space. A counte r example, ho we v er, w as found b y A. Kaplan [1], initiating the extensiv e study of g.o. s p aces [3]-[21]. Pseudo-Riemannian g.o. s p aces were also inv estigated recen tly [22, 23, 24]. Before K aplan’s example app eared, J. Szenthe disco vered a geometrica l bac kground for the situation w hen a g.o . space is not natur ally reductiv e [2], not kno wing wh ether suc h a situation can b e realized or n ot. This r esult had consid er ab le influ ence on the later studies. In general, it is p ossible th at a homogeneo us Riemannian space M = G/K is not naturally reductiv e with resp ect to G , but one can tak e other groups G ′ and K ′ so that M = G ′ /K ′ and M is naturally reductiv e with resp ect to G ′ . The same situation can o ccur for g.o. s p aces as well. It is also p ossible in some cases that a g.o. space can b e made naturally reductiv e b y taking a different symmetry group G ′ , b ut there also exist g.o. sp aces for whic h this is n ot p ossible, i.e. whic h are in no way natur ally reductiv e. Kaplan’s example is of the latter t yp e. Since Riemannian (and pseudo-Riemannian) manif olds can b e view ed as a s p ecial class of the manifolds with a L agrangia n or Hamiltonian function, it is interesting to consider the generalizat ion of the g.o. pr op ert y to homogeneous spaces with inv ariant Lagrangian and Hamiltonian functions and to ask wh ether the known r esu lts for the Riemannian spaces can b e generalized, and w hether the techniques of Lagrangian or Hamiltonian dyn amics can b e used for the study of R iemann ian g.o. spaces. In this pap er w e present the results that we obtained in r elati on to these questions. A su b ject clo sely rela ted to th e study of g.o . spaces is the c haracterization of the homo- geneous geodesics in R iemann ian manifolds. Homogeneous geod esics are of interest also in Finsler geometry , pseudo-Riemannian geometry and in dynamics. W e refer the reader to [25]-[41] an d furth er references therein. The presen t pap er is also concerned with the c haracterization of homogeneous tra jectories in Lagrangian and Hamiltonian d y n amical systems, p artly b eca u se this is necessary for the s tu dy of dynamical sy s tems th at ha v e the g.o. prop ert y . In the p h ysics literature the homogeneous geo desics are usually ca lled 2 relativ e equilibria, therefore we shall also use this term, along with the term homogeneous tra jectory . W e men tion that another name for h omoge neous geo desics that app ears in the literature is stationary geo desic. At times w e s hall use the terms Lagrangian sp ace and Hamiltonian space for Lagrangian and Hamiltonian dynamical systems, in analogy with the term Riemannian space. The pap er is organized as follo w s . In section 2 w e discuss the case of Lagrangian systems. W e describ e criteria for an orb it of a one-parameter subgroup to b e a solution of the Eu ler-Lagrange equations, including the Lagrangian v ersion of the ‘geodesic lemma’. W e also present r esults concerning the existence of relativ e equ ilibria. In sec tion 3 w e discuss the case of Ha miltonian systems. W e describ e criteria for an orbit of a one-paramete r sub group to b e a solution of the Hamiltonian equations, including th e Hamilto nian v ers ion of the geo d esic lemma. T h en w e turn to the c harac- terizatio n of Hamilt onian g.o. sp ace s. In particular, w e sho w that the Hamilto nian g.o. spaces are closely related to the fun ctions which are inv ariant u nder the co adjoin t actio n of G . Riemannian g.o. spaces corresp ond, of cour se, to s p ecial Ad ∗ ( G )-in v arian t func- tions. Naturally reductiv e metrics, in particular, are kno wn to corresp ond to quad r atic Ad ∗ ( G )-in v arian t p olynomials [42, 43 ]. An Ad ∗ ( G )-in v arian t function that is related to a g.o. space also serve s as a p oten tial for the mapping called ge o desic gr aph , whic h w as in tr o du ced originally by S zen the [2] and which h as prov ed to b e useful for the description of Riemannian g.o. spaces. W e present certain resu lts on geo desic graphs, and then w e de- scrib e a criterion based on the relatio n b et we en g.o . spaces and Ad ∗ ( G )-in v arian t functions that ca n b e used to fi nd g.o. Hamilt onians or metrics. W e also describ e a generalization of th e notion of Hamiltonian g.o. sp ace. In s ection 4 w e discuss the tw o-parameter family of Riemannian g.o. metrics on S U (3) /S U (2) for the illustration of the results of section 3. W e calcula te the geodesic graph in a new wa y , u tilizi ng the relation b et ween g.o. spaces and Ad ∗ ( G )-in v arian t func- tions. 2 Lagrangian systems with homogeneous tra jectories Let M b e a connected manifold with a Lagrangian fu nction L : T M → R on it. The Euler-L agr ange e quation for a curve γ : I → M , wh er e I is an in terv al, is ∂ L ∂ x i ( γ ( t ) , ˙ γ ( t )) = d dt  ∂ L ∂ v i ( γ , ˙ γ )  ( t ) ∀ t ∈ I , (1) or, exp anding the r igh t hand side, ∂ L ∂ x i ( γ ( t ) , ˙ γ ( t )) = ∂ 2 L ∂ x i ∂ v j ( γ ( t ) , ˙ γ ( t )) ˙ γ j ( t ) + ∂ 2 L ∂ v i ∂ v j ( γ ( t ) , ˙ γ ( t )) ¨ γ j ( t ) . (2) Here and throughout the pap er we use the Ein stein summation conv en tion for indices of co ordinates related to M . In the sp ecial case wh en L is the quadratic form corresp ondin g to a Riemannian or pseud o-Rie mannian m etric, a solution γ : I → M of the Eu ler-Lagrange equations is a geodesic with affine p arametrizat ion. 3 The Lagrangia n is r e gular if the bilinear form ∂ 2 L ∂ v i ∂ v j ( x, v ) is nondegenerate for any ( x, v ) ∈ T M . The regularit y of a Lagrangia n implies that the solution of the Euler- Lagrange equations is u nique for giv en initial data ( x, v ) ∈ T M . If a Lagrangian corre- sp onds to a metric, then it is regular. In the foll o wing w e assume that L is inv ariant u n der the action of a connected Lie group G on T M induced by an action of G on M . W e denote the Lie deriv ativ e with r esp ect to a vect or field Z as L Z . W e use the notation ◦ for the comp osition of t wo f unctions, i.e. if f and g are t wo f u nctions, then f ◦ g is the function f or whic h ( f ◦ g )( x ) = f ( g ( x )). In the deriv ation of the results of th is s ection the Euler-Lagrange equati on, an equation expressing the inv ariance of L and equations characte rizing the v elo cit y and acceleratio n of orb its ha ve imp ortant role. Let Z a : M → T M an d ˆ Z a : T M → T T M , where a ∈ g , b e the infinitesimal generator v ector fields for th e action of G on M and T M , resp ectiv ely . Their coord inate form is Z a ( x ) = ∂ φ i a ∂ τ (0 , x ) ∂ ∂ x i , x ∈ M (3) and ˆ Z a ( x, v ) = ∂ φ i a ∂ τ (0 , x ) ∂ ∂ x i + ∂ 2 φ i a ∂ τ ∂ x j (0 , x ) v j ∂ ∂ v i , ( x, v ) ∈ T M , (4) where φ a : R × M → M is the action of the one-parameter subgroup generated by a ∈ g and τ denotes the firs t v ariable of φ a . The inv ariance of L un d er the action of G implies the follo wing symmetry c ondition : L ˆ Z a L ( x, v ) = ∂ L ∂ x i ( x, v ) ∂ φ i a ∂ τ (0 , x ) + ∂ L ∂ v i ( x, v ) ∂ 2 φ i a ∂ τ ∂ x j (0 , x ) v j = 0 , (5) where a ∈ g . T h is equation holds for all ( x, v ) ∈ T M . The orbit of the one-parameter s ubgroup generated by a ∈ g in M with initial p oin t x is the curve γ : I → M , t 7→ φ a ( t, x ). F o r the velocit y ˙ γ ( t ) = ∂ φ a ∂ τ ( t, x ) (6) of th is orbit the equation ˙ γ i ( t ) = ∂ φ i a ∂ x j ( t, x ) ˙ γ j (0) = ∂ φ i a ∂ x j ( t, x ) ∂ φ j a ∂ τ (0 , x ) (7) holds b eca use of the group pr op erty . F or th e accelerat ion w e h a v e ¨ γ i ( t ) = ∂ 2 φ i a ∂ τ ∂ x j ( t, x ) ∂ φ j a ∂ τ (0 , x ) = ∂ 2 φ i a ∂ τ 2 ( t, x ) . (8) 4 Theorem 2.1 The orbit of a one-p ar ameter sub gr oup of G starting at x ∈ M is a solution of the E uler-L agr ange e quations c orr esp onding to the (not ne c essarily r e gular) L agr angian L if and only if x is a critic al p oint of the function L ◦ Z a , i.e. d ( L ◦ Z a )( x ) = 0 , (9) wher e Z a is the gener ator ve c tor field of the sub gr oup. Pro of. Because of the in v ariance of the Lagrangian an orbit of a one-parameter symmetry group is a solution of the Eu ler-Lagrange equations if and only if it satisfies the Euler- Lagrange equ ations at the initial p oin t. First, let u s assume that the orbit is a solution of the Eu ler-Lagrange equations. Differentia ting the symmetry condition (5) with resp ect to v j yields 0 = ∂ ∂ v j L ˆ Z a L ( x, v ) = ∂ 2 L ∂ x i ∂ v j ( x, v ) ∂ φ i a ∂ τ (0 , x ) + ∂ 2 L ∂ v i ∂ v j ( x, v ) ∂ 2 φ i a ∂ τ ∂ x k (0 , x ) v k + ∂ L ∂ v i ( x, v ) ∂ 2 φ i a ∂ τ ∂ x j (0 , x ) . (10) Substituting th e righ t h and side of (8) for ¨ γ in the Euler-Lagrange equation (2) at t = 0 giv es ∂ L ∂ x j ( x, v ) = ∂ 2 L ∂ x i ∂ v j ( x, v ) ∂ φ i a ∂ τ (0 , x ) + ∂ 2 L ∂ v i ∂ v j ( x, v ) ∂ 2 φ i a ∂ τ ∂ x j (0 , x ) ∂ φ j a ∂ τ (0 , x ) , (11) where v = ˙ γ (0). Setting v = ˙ γ (0) also in (10) and subtracting fr om (11) giv es ∂ L ∂ x j ( x, v ) + ∂ L ∂ v i ( x, v ) ∂ 2 φ i a ∂ τ ∂ x j (0 , x ) = 0 , (12) where v = ˙ γ (0), wh ic h is just the coordinate form of (9). Considering the rev erse direction of th e statemen t, it is clear now that if (12) and (10) hold, then (11) follo ws. ✷ A similar theorem is stated in [31] (see also [41]). Ho w ever, our pro of is differen t from those giv en in [31] and [41]. The function L ◦ Z a is call ed augmente d L agr angian in [31] and lo cke d L agr angian in [41]. Definition 2.2 An elemen t a of g is called a r elative e qui librium ve ctor at x ∈ M if the orbit of the one-parameter subgroup of G generated by a and starting at x is a solution of th e Euler-Lagrange equations. In Riemannian geometry the in teresting rela tiv e equilibrium v ectors are, of course, those whic h generate orbits that are not single p oin ts in M . W e n ote that in Riemannian geometry the relativ e equilibrium v ectors are us ually called ge o desic ve ctors . The set of relativ e equilibrium vec tors at x is in v ariant u nder G x , the stabilizer of x . If g x = y for some x, y ∈ M and g ∈ G , then the set of r elat iv e equilibriu m ve ctors at y can b e obtained from that at x by the adjoin t action of g . 5 As regards the existence of relati v e equilibria, the follo wing coroll ary of theorem 2.1 can b e stated. Theorem 2.3 L et M , G , L b e as in the the or em 2.1 and let M b e c omp act. F or any a ∈ g ther e exists at le ast one solution of the Euler-L agr ange e quations which is the or- bit of the o ne-p ar ameter sub gr oup gener ate d by a . If ther e exists an a ∈ g such that Z a ( x ) 6 = 0 ∀ x ∈ M , then ther e exists at le ast one solution of the Euler-L agr ange e quations which is the orbit of the one-p ar ameter sub gr oup gener ate d by a and is not a single p oint in M . If, in addition, M is also hom o ge ne ous with r esp e c t to the action of G , then ther e exists at le ast one nonzer o r elative e quilibrium ve ctor at every p oint in M , which gener ates an orbit that is not a single p oint. This result can b e found e.g. in [27] (prop osition 5.2) for the sp ecial case of Lagrangians that d escrib e geod esic motion in Riemannian manifolds. In the r est of this section we consider the case when M is a h omoge neous s p ace. F or a h omogeneous space M = G/K there is a linear map f x : g → T x M , a 7→ Z a ( x ) for eac h p oint x ∈ M . W e use the notation f for f o (i.e. we omit the sub s cript o denoting the origin in G/K ). The dual of a v ector space V w ill b e d enoted by V ∗ . The con traction (or natural pairing) b et w een V and V ∗ will b e denoted in the follo wing w ay: ( w | v ), w here w ∈ V ∗ and v ∈ V . Th e transp ose of a linear map A : V → W will b e d en oted b y A ∗ (it is d efined as A ∗ : W ∗ → V ∗ , w 7→ w ◦ A ). The follo w ing lemma, whic h concerns homogeneous manifolds with inv ariant Lagrang- ians and is the generalization of the known ‘geo desic lemma’ for the Riemannian case [6] (see also for example [33 , 7, 9]), gives a condition for an element of g to b e a relativ e equilibrium vec tor at o . This is a local condition in the sense th at it is giv en in terms of L restricted to T o M , the elemen ts of g , and the v alues of the infi nitesimal generator v ector fields at o . In Riemannian geometry the geodesic lemma has p ro v ed to b e v ery u seful in the study of homogeneous geo desics. Lemma 2.4 (Geo desic lemma) L et M = G/K b e a homo g ene ous sp ac e with a G - invariant L agr angian L : T M → R . An element a ∈ g is a r elative e quilibriu m v e ctor at o if and only if ( dL o ( f ( a )) | f ([ a, b ]) ) = 0 ∀ b ∈ g , (13) wher e L o is L r estricte d to T o M . In p articular, if L c orr esp onds to a Riema nnian metric, then (13) takes the form h f ([ a, b ]) , f ( a ) i = 0 ∀ b ∈ g , (14) or, e quivalently, h [ a, b ] m , a m i = 0 ∀ b ∈ g , (15) wher e the index m denotes the m -c omp onent r elate d to a r e ductive de c omp osition g = k ⊕ m , and m is assume d to b e identifie d with T o M by f . 6 Pro of. Let us assume fir st, that a is a relati v e equilibrium vec tor. (9) in theorem 2.1 is equiv alen t to L Z b ( L ◦ Z a )( o ) = 0 ∀ b ∈ g . In co ord inate form L Z b ( L ◦ Z a )( o ) = ∂ φ i b ∂ τ (0 , o ) ∂ L ∂ x i ( o, ∂ φ a ∂ τ (0 , o )) + ∂ φ i b ∂ τ (0 , o ) ∂ L ∂ v j ( o, ∂ φ a ∂ τ (0 , o )) ∂ 2 φ j a ∂ τ ∂ x i (0 , o ) = 0 . (16) T aking the symmetry condition (5) at th e p oint ( o, ∂ φ a ∂ τ (0 , o )) we get L ˆ Z b L ( o, ∂ φ a ∂ τ (0 , o )) = ∂ φ i b ∂ τ (0 , o ) ∂ L ∂ x i ( o, ∂ φ a ∂ τ (0 , o )) + ∂ L ∂ v i ( o, ∂ φ a ∂ τ (0 , o )) ∂ 2 φ i b ∂ τ ∂ x j (0 , o ) ∂ φ j a ∂ τ (0 , o ) = 0 . (17) Subtracting th ese t w o equatio ns giv es ∂ L ∂ v j ( o, ∂ φ a ∂ τ (0 , o )) " ∂ φ i b ∂ τ (0 , o ) ∂ 2 φ j a ∂ τ ∂ x i (0 , o ) − ∂ φ i a ∂ τ (0 , o ) ∂ 2 φ j b ∂ τ ∂ x i (0 , o ) # = 0 , (18) whic h is the co ordinate expression for (13). Con versely , assu ming that (18) holds and using (17) one obtains (16). The second part of the lemma concerning the Riemannian case follo w s ob viously from the fir st part. ✷ The formula (15) f or Riemannian spaces is well kno wn and is also a generalization of Arnold’s r esult ab out homogeneous geo d esics of left-in v ariant metrics on Lie grou p s [26]. Let r : R → g b e the adjoint orbit starting at a and generated by b . f ([ a, b ]) is the tangen t v ector of the curv e f ◦ r at the point f ( a ). Equation (13) means that the d eriv ativ e of L o at f ( a ) along this tangent ve ctor is 0. The follo wing theorems 2.5 and 2.6 are ab out th e existence of relativ e equilibria. Theorem 2.5 L et M = G/ K b e a homo gene ous sp ac e with a G -inv ariant L agr angian L : T M → R . If G i s c omp act, then e ach adjoint orbit of G c ontains at le ast one r elative e quilibrium ve ctor at o , and e ach adjoint orbit of G that is not c ontaine d entir ely by k c ontains at le ast one r elative e quilibrium ve ctor at o which gener ates an orbit that is not a single p oint. Pro of. Any adjoint orbit O of G is compact. f ( O ) is also compact and L o is con tin u ous on it, th u s there exists at least one ˜ v ∈ f ( O ) so that L o | f ( O ) is minimal or maximal at ˜ v . Because of this extremalit y the deriv ativ e of L o is zero at ˜ v along an y curv e that lies in f ( O ) and passes through ˜ v . It is clear fr om the remark after the pr oof of the geod esic lemma that any elemen t of f − 1 ( ˜ v ) ∩ O is a r elat iv e equilibrium ve ctor at o . If an adjoint orbit O is not con tained en tirely b y k , then f ( O ) 6 = { 0 } , th us there exists at least one ˜ v ∈ f ( O ) so that ˜ v 6 = 0 and L o | f ( O ) is minimal or maximal at ˜ v . An y elemen t 7 of f − 1 ( ˜ v ) ∩ O is a relativ e equilibr ium v ector at o that generates an orb it that is not a single p oin t. ✷ Theorem 2.6 L et M = G/ K b e a homo gene ous sp ac e with a G -inv ariant L agr angian L : T M → R . If G is solvable and the image sp ac e of dL o | T o M \{ 0 } c ontains ve ctors of arbitr ary dir e ction, than ther e exists at le ast one r elative e quilibrium ve ctor at o , which gener ates an orbit that is not a single p oint. Pro of. Consider the d eriv ed series of g , i.e. th e sequence g (0) ⊃ g (1) ⊃ · · · ⊃ g ( i ) ⊃ . . . , where g (0) = g and g ( i ) = [ g ( i − 1) , g ( i − 1) ] for i = 1 , 2 , . . . . Because of the solv abilit y of G , the deriv ed series strictly decreases and ends in the null space. Consequen tly , there exists an ind ex r ≥ 0 su c h that f ( g ( r ) ) = T o M , but f ( g ( r +1) ) is a prop er su bspace of T o M . The connected subgroup G ( r ) corresp onding to g ( r ) still acts transitiv ely on M , therefore it is necessary and sufficien t for a vec tor to b e a relativ e equilibrium v ector that (13) hold for all b ∈ g ( r ) . Th e co ndition imp osed on dL o in the theorem ensures that there exists an ˜ v ∈ T o M \ { 0 } suc h that ( dL 0 ( ˜ v ) | f ([ g ( r ) , g ( r ) ]) ) = 0, implying that any elemen t of f − 1 ( ˜ v ) ∩ g ( r ) is a relativ e equilibriu m v ector. ✷ This theorem is similar to some parts of pr op ositio n 3 of [33]. It is clea r fr om the pro of that the s olv abilit y of G is not necessary , it can b e replaced by the we ak er condition that there exists an elemen t g ( r +1) of the deriv ed series of g s uc h that f ( g ( r +1) ) is a prop er subspace of T o M . The condition of regularit y has not b een imp osed on th e Lagrangians so far. It is assumed, ho wev er, in the follo wing t wo prop ositions 2.8 and 2.10, wh ic h c h aracterize La- grangian g.o. spaces. Definition 2.7 Let M = G/K b e a homogeneous sp ace and let L : M → R b e a G - in v ariant Lagrangian f unction. ( M , L ) is a called a L agr angian ge o desic orbit (g.o.) sp ac e with r esp e ct to G , if ev ery solution of the Eu ler-Lagrange equations corresp onding to L is an orbit of a one-parameter sub group of G . In other w ords , a Lagrangian g.o. sp ace is defined by the prop ert y that ev ery solution of the Euler-Lagrange equ ations is a relativ e equilibrium. In the general Lagrangian me- c hanical con text one could in tro duce a new name instead of ‘geo desic orbit space’, since the latt er b ears a reference to Riemannian geometry . I n the present pap er, ho we v er, w e shall not in tr od u ce su c h a name. This applies also to the ’geodesic lemma’ and to the ‘geod esic graph ’ defined b elo w. 8 Prop osition 2.8 L et M = G/K and L b e as in definition 2.7, and assume that L is r e gu lar. The L agr angian dyna mic al system ( M , L ) has the g.o. pr op erty with r esp e ct to G if and only if for al l v ∈ T o M ther e exists an a ∈ g such that f ( a ) = v and a i s a r elative e quilibrium ve ctor. Definition 2.9 Let ( M = G/K, L ) b e a L agrangia n system that has the g.o . prop erty with resp ect to G . A mapping ξ : T o M → g with the prop erties that f ( ξ ( v )) = v and ξ ( v ) is a relativ e equilibrium v ector at o for all v ∈ T o M is called a ge o desic gr aph . Obviously , there exists at least one geo desic graph for ev ery Lagrangian system that has the g.o. prop ert y . f ( ξ ( v )) = v means that the vel o cit y of the orbit generated by ξ ( v ) is v at o . In Riemannian geo metry th e geo d esic graph is v ery useful for studyin g g.o. spaces. Imp ortan t results ab out its prop erties were obtained in [2, 9]. It follo w s d irectly from the definition of naturally reductiv e metrics in the In tro du ction and from 2.9 that the naturally reductiv e sp aces are p recisely those Riemannian g.o. spaces that admit a K -equiv arian t linear geodesic graph. In order to see this in detail, assume fi rst that M = G/K is a natur ally reductive space with the natural r eductiv e decomp osition g = k ⊕ m . Then f | m is a linear bijection b et we en m and T o M , and its in verse ξ = ( f | m ) − 1 ob viously has the prop erty f ( ξ ( v )) = v . ξ is also K -equiv ariant , since m is an Ad ( K )-in v arian t s u bspace of g . The natur al redu ctivit y of M implies that for an y v ∈ T o M there is an a ∈ m so that the orbit generated b y a and starting at o coincides w ith the geodesic with initial ve lo cit y v . Ho w ever, the initial v elo city of the orbit generated by a is f ( a ), therefore a = ξ ( v ). T his sho ws that ξ is a K -equiv ariant linear geod esic graph. Conv ersely , if M is a Riemannian g.o. sp ace and ξ is a K -equiv ariant linear geo desic graph, then ξ ( T o M ) is an Ad ( K )-inv arian t linear su bspace of g , d ue to the linearit y and K -equiv ariance of ξ . ξ ( T o M ) is complemen tary to k , b ecause f ( k ) = 0 and f ( ξ ( T o M )) = T o M . g = k ⊕ ξ ( T o M ) is thus a reductiv e decomp osition. By defin itio n 2.9, for an arbitrary geo desic γ starting at o the Lie algebra elemen t ξ ( v ) ∈ ξ ( T o M ), wh er e v is the initial v elocity of γ , generates an orbit ˜ γ that is also a geo desic with initial v elo cit y v . Since geodesics are uniquely determined by their initial data, ˜ γ and γ coincide. This sho w s that the r eductiv e decomp osition g = k ⊕ ξ ( T o M ) is also natural. W e note that there is a minor difference b et w een our definition of th e ge o desic graph and the u sual definition; in th e usual definition one has a direct su m decomp osition g = m ⊕ k , and one tak es the k -comp onent of ξ ( v ) as the v alue of the geodesic graph at v , since the m -comp onen t is uniquely determin ed by the prop erty f ( ξ ( v )) = v . In fact, in the literature m is often iden tified with T o M b y f . It is also usual in the literature to include in the definition of the geod esic graph the requiremen t that it s hould b e K - equiv arian t. The follo win g consequence of prop osition 2.8 and of the geo desic lemma, in particular of (13), applying to the sp ecial case M = G , is wel l kno wn [28]. Theorem 2.10 If M = G , i.e. L is a r e gular left- invariant L agr angian on G , then ( M , L ) is a g.o. sp ac e with r esp e ct to G if and only i f L e = L | T e G (wher e e is the unit element 9 of G ) is invariant under the adjoint action of G . Any function on T e G c an b e extende d uniquely to a left- invariant function on G , ther efor e the L agr angians on G that have the g.o. pr op erty with r e sp e ct to G ar e in one-to-one c orr e sp ondenc e with the r e g ular Ad - invariant func tions on g . W e note that in the case M = G the equation (13) expresses the Ad ( G )-in v ariance of L e . In the n ext section we turn to the Hamilt onian formalism, w h ic h is b etter suited to the c haracterization of g.o. spaces than the Lagrangian formalism. 3 Hamiltonian systems with homogeneous tra jectories Let M b e a manifold w ith a Hamiltonian function H : T ∗ M → R . W e denote the Hamilto - nian v ector fi eld generated by H on the symplectic manifold T ∗ M by X H . In co ordinates X H is giv en b y X H ( x, p ) =  ∂ H ∂ p i ( x, p ) , − ∂ H ∂ x i ( x, p )  . The Hamiltonian e quations for a curv e γ : I → T ∗ M are the f ollo wing: X H ( γ ( t )) = ˙ γ ( t ) ∀ t ∈ I , (19) or equiv alen tly ∂ H ∂ p i ( x, p ) = ˙ x i (20) − ∂ H ∂ x i ( x, p ) = ˙ p i . (21) The pro jection of a solution γ : I → T ∗ M on M is a geod esic with affine parametrization in the sp ecial case w hen H is the quadratic form corresp onding to a Riemannian or pseud o- Riemannian metric. In the follo w ing w e assume that H is inv arian t under the action of a connected Lie group G on T ∗ M indu ced by an action of G on M . Let ˆ Z ∗ a : T ∗ M → T T ∗ M , a ∈ g , b e the infinitesimal generator v ector fields for the action of G on T ∗ M . Th eir co ordinate form is ˆ Z ∗ a ( x, p ) = ∂ φ i a ∂ τ (0 , x ) ∂ ∂ x i − ∂ 2 φ j a ∂ τ ∂ x i (0 , x ) p j ∂ ∂ p i , (22) where φ a is the same ob j ect as in section 2. The inv ariance of H implies the follo wing symmetry c ondition : L ˆ Z ∗ b H ( x, p ) = ∂ H ∂ x i ( x, p ) ∂ φ i b ∂ τ (0 , x ) − ∂ H ∂ p i ( x, p ) ∂ 2 φ j b ∂ τ ∂ x i (0 , x ) p j = 0 , (23) where b ∈ g . T h is equation holds for all ( x, p ) ∈ T ∗ M . W e recall that the momentum map for the action of G on T ∗ M is P : T ∗ M → g ∗ , ( x, p ) 7→ f ∗ x ( p ), wher e f x is the linear mappin g in tro duced in sectio n 2 after theorem 10 2.3. Clea rly P is linear on eac h cotangen t space T ∗ x M , x ∈ M , and it is also equiv arian t. P restricted to the cotangen t space T ∗ x M at x ∈ M is the tran s p ose of f x . P h as th e prop ert y that X ( P | a ) = ˆ Z ∗ a ∀ a ∈ g , (24) where X F denotes the Hamiltonian v ector field corresp onding to the function F : T ∗ M → R and ( P | a ) denotes the fu nction ( x, p ) 7→ ( f ∗ x ( p ) | a ). This p rop ert y implies [ X ( P | a ) , X ( P | b ) ] = X ( P | [ a,b ]) , where [ , ] on the le ft hand side denotes the Lie b rac k et of v ector fields. T he f unc- tions ( P | a ), a ∈ g , are conserv ed quan tities, i.e. the function P (and thus ( P | a ), for all a ∈ g ) is constan t along the solutions of the Hamiltonian equations. Definition 3.1 An elemen t a of g is called a r elative e quilibrium ve ctor at ( x, p ) ∈ T ∗ M if th e orbit of the corresp ond ing one-paramete r su bgroup starting at ( x, p ) is a solution of the Hamiltonian equations. Since the momen tum map is constan t along the solutions of the Hamilto nian equa- tions, if a ∈ g is a relativ e equilibrium ve ctor at ( x, p ) ∈ T ∗ M , then a is an elemen t of the stabilizer subgroup of P ( x, p ) with r esp ect to th e coadjoin t action of G . Lemma 3.2 L e t H : T ∗ M → R b e a Hamiltonian function that is invariant under the action of a c onne cte d Lie gr oup G . a ∈ g is a r elative e quilibrium v e ctor at ( x, p ) ∈ T ∗ M if and only if X H ( x, p ) = ˆ Z ∗ a ( x, p ) , (25) or, e quivalently, d ( H − ( P | a ))( x, p ) = 0 , (26) wher e P is the momentum mapping for the action of G on T ∗ M . The p ro of of this lemma can b e found in [28] (pr op osition 4.3.7.), for ins tance. The follo wing generalizatio n of the geodesic lemma can b e stated for homogeneous spaces w ith in v arian t Hamiltonians. Lemma 3.3 (Geo desic lemma) L et M = G/K b e a homo gene ous sp ac e and H : T ∗ M → R a G -invariant Hamiltonian function. An element a ∈ g is a r elative e qui- librium ve ctor at ( o, p ) , wher e o denotes the origin, if and only if dH o ( p ) = f ( a ) (27) and ( f ∗ ( p ) | [ a, b ] ) = 0 ∀ b ∈ g (28) hold, wher e H o is H r e stricte d to T ∗ o M . (28) is e quivalent to the c ondition that the one- p ar ameter sub gr oup gener ate d by a is c ontaine d by the stabilizer sub gr oup of f ∗ ( p ) ∈ g ∗ with r esp e ct to the c o adjoint action of G . 11 Pro of. Assume first that a is a r elat iv e equilibrium v ector. (27) is just the fi rst of the t wo Hamiltonian equations at the initial p oint and in co ordinate form it r eads as follo ws: ∂ φ i a ∂ τ ( x, 0) = ∂ H ∂ p i ( x, p ) . (29) The second Hamiltonian equation at the initial p oin t is ∂ 2 φ j a ∂ τ ∂ x i (0 , x ) p j = ∂ H ∂ x i ( x, p ) . (30) Substituting the left hand sides of (29) and (30) for the r igh t hand sides of (29) and (30) in (23) giv es p j " ∂ φ i a ∂ τ (0 , x ) ∂ 2 φ j b ∂ τ ∂ x i (0 , x ) − ∂ φ i b ∂ τ (0 , x ) ∂ 2 φ j a ∂ τ ∂ x i (0 , x ) # = 0 ∀ b ∈ g , (31) whic h is just the co ordinate form of the equation ( p | [ Z a , Z b ]( o ) ) = 0 ∀ b ∈ g . (32) This is equiv alen t to (28), b ecause [ Z a , Z b ]( o ) = f ([ a, b ]) and ( p | f ([ a, b ])) = ( f ∗ ( p ) | [ a, b ]). Considering th e rev erse direction, it is clear that (30) can b e obtained from (31 ), (29) and (23). ✷ Prop osition 3.4 The set of r elative e qui librium ve ctors at any p oint ( o, p ) is an affine subsp ac e of g . Pro of. F or an y fix ed p the equations (27) and (28) constitute an inhomogeneous linear system of equations for a , thus the solutions constitute an affine sub space in g . ✷ A similar result holds f or Lagrangian systems as w ell; in th is case the statemen t is that the set of relati v e equilibrium ve ctors a at o f or which f ( a ) (whic h is the initial v elo cit y of th e orbit generated by a ) is fixed is an affine subspace of g . This follo ws from the fact that the equ atio ns f ( a ) = v and (13), where v is fixed, constitute an inhomogeneous linear system for a . Definition 3.5 Let M = G/K a homogeneous space and H : T ∗ M → R a G -in v ariant Hamiltonian function. ( M , H ) is a called a Hamilton ian ge o desic orbit (g.o.) sp ac e with r e sp e ct to G , if every solution of the Hamiltonian equations is an orbit of a one-parameter subgroup of G . In the follo wing prop ositions 3.6 and 3.7 ele men tary co nditions are given under whic h a homogeneous space with an inv ariant Hamiltonian has the g.o. p rop ert y . They are dir ect consequences of lemma 3.2 and lemma 3.3. 12 Prop osition 3.6 L et M = G/K b e a homo gene ous sp ac e and H : T ∗ M → R a G - invariant Hamiltonian function. This dynamic al system has the g.o. pr op erty with r esp e c t to G i f and only if dH ( o, p ) ∈ { d ( P | b )( o, p ) : b ∈ g } ∀ ( o, p ) ∈ T ∗ o M (33) or, e quivalently, X H ( o, p ) ∈ { ˆ Z ∗ b ( o, p ) : b ∈ g } ∀ ( o, p ) ∈ T ∗ o M . (34) Prop osition 3.7 L et M and H b e the same as in the pr evious pr op osition. ( M , H ) is a g.o. sp ac e with r e sp e ct to G if and only if for al l p ∈ T ∗ o M ther e exi sts an a ∈ g such tha t dH o ( p ) = f ( a ) (35) and ( f ∗ ( p ) | [ a, b ] ) = 0 ∀ b ∈ g (36) hold. Definition 3.8 Let M = G/K b e a Hamiltonia n g.o. space with resp ect to G . A mappin g ξ : T ∗ o M → g with the prop erty that ξ ( p ) is a relativ e equilibrium vec tor at ( o, p ) for all p ∈ T ∗ o M is called a ge o desic gr aph . Obviously , there exists at least one geo desic graph for ev ery Hamiltonian g.o. space. The naturally reductiv e spaces are precisely those Riemannian g.o. spaces whic h admit a linea r K -equiv ariant geo desic graph. If M = G/K is naturally reductiv e and g = k ⊕ m is a natural redu ctiv e d ecomposition, then the mappin g ξ defin ed as ξ ( p ) = ( f | m ) − 1 ( dH o ( p )) is a linear K -equiv ariant geodesic graph. The mapping p 7→ dH o ( p ) is a linear b ijectio n b et ween T ∗ o M and T o M in this case, since H o is quadratic an d nondegenerate. If M is a Riemannian g.o. space and ξ is a linear K -equiv ariant geo desic graph , then g = k ⊕ ξ ( T ∗ o M ) is a natural reductiv e d eco mp osition. (See also the r emarks after definition 2.9.) In the follo wing last part of the section we describ e the relation b et ween g.o. spaces and Ad ∗ ( G )-in v arian t functions, and w e d escrib e ho w an Ad ∗ ( G )-in v arian t function that corresp onds to a g.o. space can b e used to obtain a geo desic graph. W e present certain results on geo desic graphs and we d iscuss Riemannian g.o. spaces and natur ally reductiv e spaces. W e also describ e a criterion that can b e used to find Hamiltonians or m etrics that h av e th e g.o. prop erty . Finally , w e d iscuss briefly a generaliz ation of the n otion of Hamiltonian g.o. space. 13 Lemma 3.9 L et M = G/K b e a homo gene ous sp ac e and H : T ∗ M → R a G -invariant Hamiltonian function that has the g.o. pr op erty with r esp e ct to G . If P is c onstant alo ng a smo oth curve γ : I → T ∗ M , then H is also c onstant along this cu rve. Pro of. T he der iv ativ e d ( H ◦ γ ) dt of H along γ at t ∈ I equals ( dH ( γ ( t )) | ˙ γ ( t )). It is sufficient to sh o w that th is num b er is zero for an y t ∈ I . Let t b e a fixed element of I . It follo w s from pr op osition 3.6. th at ( dH ( γ ( t )) | ˙ γ ( t )) = ( d ( P | b )( γ ( t )) | ˙ γ ( t )) for some b ∈ g . Since P is constant along γ , the deriv ativ e of P alo ng γ is zero, therefore the deriv ativ e of ( P | b ) is also zero, thus ( d ( P | b )( γ ( t )) | ˙ γ ( t )) = 0. ✷ The follo wing theorem is a direct consequence of lemma 3.9. Theorem 3.10 L et M = G/ K b e a homo ge ne ous sp ac e and H : T ∗ M → R a G -invariant Hamiltonian function that has the g.o. pr op erty with r esp e ct to G . If the c onne cte d c omp o- nents of the level sets of the momentum mapping P have the pr op erty that any two p oint in them c an b e c onne c te d by a pie c ewise smo oth curve, then H is c onstant on the c onne cte d c omp onents of the level sets of P . If, in addition, H takes the same value on al l c onne cte d c omp onents of any level set of P , then H takes the form H = h ◦ P , (37) wher e h : g ∗ → R is an Ad ∗ ( G ) -invariant function. P is an analytic function, therefore its rank is maximal on an op en d ense subset N of T ∗ M , whic h is G -inv ariant . It follo ws that in N th e lev el sets of P are submanifolds, therefore the co ndition of theorem 3.10 is satisfied and thus H is constan t on th e connected comp onen ts of the leve l sets of P | N . The form ula H = h ◦ P alw a ys holds locally in N ; if ( o, p ) is in N , then there exists a suitable op en neigh b orh oo d O of ( o, p ) in N so that in this neigh b ourho od H tak es the form H = h ◦ P , where h is a (lo cally) Ad ∗ ( G )-in v arian t smo oth fun ctio n on P ( O ). F ur thermore, it follo ws from the pro of of theorem 3.11, that if dim P ( O ) = dim G , then ξ : p ′ 7→ dh ( P ( o, p ′ )) is a smo oth (locally) K -equiv ariant geodesic graph in an op en neigh b ou r ho od of p in T ∗ o M . If dim P ( O ) < dim G , then h can b e extended to an op en neigh b orh oo d of P ( O ), and this extended v ersion can b e us ed to define ξ . If the extension ˆ h of h is K -in v arian t, then the lo cal geodesic graph giv en b y ξ : p ′ 7→ d ˆ h ( P ( o, p ′ )) is also K -equiv arian t. The follo w in g theorem is a conv erse of theorem 3.10. Summation o v er the in dex n is implied in the form ulas (39) and (41). Theorem 3.11 L et h : g ∗ → R b e an Ad ∗ ( G ) -invariant function with the pr op erties that h ◦ f ∗ is smo oth and h is differ entiable at the p oints of the image sp ac e of f ∗ (which i s f ∗ ( T ∗ o M ) ). The Hamiltonian function define d as H = h ◦ P (38) 14 is G -invariant and has the g.o. pr op erty. The ve ctor dh ( P ( o, p )) = ∂ h ∂ g n ( P ( o, p )) dg n , (39) wher e the g n ar e some line ar c o or dinates on g ∗ , is a r elative e quilibriu m ve ctor at ( o, p ) ∈ T ∗ M , thus the mapping ξ = dh ◦ f ∗ : T ∗ o M → g , p 7→ dh ( P ( o, p )) ≡ ( dh ◦ f ∗ )( p ) (40) is a K -e quivariant ge o desic gr aph. Pro of. W e note that P ( o, p ) = f ∗ ( p ), b y d efinition. H is ob viously G -in v arian t. Th e prop ert y that h ◦ f ∗ is smo oth implies the smo othness of H . W e h a v e dH = ∂ h ∂ g n ∂ P n ∂ x j dx j + ∂ h ∂ g n ∂ P n ∂ p j dp j , (41) where P n are the co mp onent s of P with resp ect to the coord inates g n . This sh o ws that at ( o, p ) ∈ T ∗ M the vect or b ∈ g th at has the comp on ents ∂ h ∂ g n ( P ( o, p )) has the p rop ert y that dH ( o, p ) = d ( P | b )( o, p ), th us the condition of pr op osition 3.6 is fu lfilled. Clearly ∂ h ∂ g n ( P ( o, p )) are just the comp onen ts of dh ( P ( o, p )) w ith resp ect to the coord inates g n . ✷ It is also clear fr om the pr oof of theorem 3.11 that Prop osition 3.12 If h : g ∗ → R is an Ad ∗ ( G ) -invariant function, H = h ◦ P is a smo oth Hamiltonian function and h is differ entiable at P ( o, p ) for some p ∈ T ∗ o M , then dh ( P ( o, p )) is a r elative e qui librium ve ctor at ( o, p ) . The co ndition imp osed on h in theorem 3.11 could probably b e w eak ened , in p articular w e d o not exp ect that the differen tiabilit y of h in ev ery p oin t of f ∗ ( T ∗ o M ) is necessary f or h ◦ P to b e a g.o. Hamiltonian. The foll o wing pr op ositio ns 3. 13-3.1 6, theorem 3. 18, and p artly theorem 3.17 , are ab out Riemannian and pseud o-Rie mannian spaces. Prop osition 3.13 If h i s a quadr atic Ad ∗ ( G ) -invariant p olynomial on g ∗ and the p olyno- mial h ◦ f ∗ is homo gene ous, quadr atic and nonde g e ner ate, then h gives rise to a R iemannian or pseudo-Riemannian g.o. metric on M = G/K . On T ∗ o M the quadr atic p olynomial that c orr esp onds to the metric is h ◦ f ∗ . The ge o desic gr aph ξ : p 7→ dh ( P ( o, p )) is line ar in this c ase. If h ◦ f ∗ is p ositive definite, then the metric is natur al ly r e ductive. Pro of. Th e Hamiltonian H = h ◦ P restricted to T ∗ o M is h ◦ f ∗ , and the latt er is a non- degenerate h omoge neous quadratic p olynomial, therefore H corresp onds to a Riemann ian or pseudo-Riemannian metric. The quadraticit y of h imp lies that dh is linear. P ( x, p ) is also lin ear in the second v ariable, therefore ξ : p 7→ dh ( P ( o, p )) is a lin ear map. If h ◦ f ∗ is p ositiv e defi n ite, then the corresp ondin g metric on M is Riemannian. The linearit y (and 15 the K -equiv ariance) of ξ implies, according to the remarks after definitions 2.9 and 3.8, that the metric is also naturally reductiv e. ✷ Prop osition 3.14 If h is a smo oth Ad ∗ ( G ) -invariant function on g ∗ and h ◦ f ∗ is a homo- gene ous p ositive definite quadr atic p olynomial, then h defines a nat ur al ly r e ductive sp ac e. Pro of. h giv es rise to a Riemannian metric, sin ce h ◦ f ∗ is a h omogeneous p ositiv e defin ite quadratic p olynomial. h is smo oth, therefore we can tak e its quadratic part h (2) at 0 ∈ g ∗ . h (2) is defin ed as h (2) ( a ) = 1 2 P n,m ∂ 2 h ∂ g n ∂ g m (0) a n a m , where g n are linear co ordinates on g ∗ , a ∈ g ∗ , and a n are the comp onen ts of a with resp ect to the co ordinates g n . h is Ad ∗ ( G )- in v ariant and the action of Ad ∗ ( G ) is linear, therefore ∂ 2 h ∂ g n ∂ g m (0), as an elemen t of g ⊗ g , is a G -in v ariant tensor, and th u s h (2) is also Ad ∗ ( G )-in v arian t. Moreo v er, h ◦ f ∗ = h (2) ◦ f ∗ , since f ∗ is linear and injectiv e, thus h (2) giv es rise to the same m etric as h . As a conse- quence, ξ : p 7→ dh (2) ( P ( o, p )) is a K -equiv arian t linear geod esic graph, implying that the metric d efined b y h is naturally red u ctiv e. ✷ The p ositiv e definiteness of h ◦ f ∗ is not essentia l in the pro of of th is p rop osition; it is needed only to ensur e that the metric to which h giv es rise is p ositiv e definite. Th e condi- tion that h is smo oth can al so b e relaxe d to the cond ition that h is t w ice differentia ble at 0. F rom a theorem of K ostant [42] ge neralized b y D’A tri and Ziller [43] it also follo ws that all naturally reductive metrics can b e obtained from h functions that are nondegenerate (not necessarily p ositiv e definite) qu adratic p olynomials. More sp ecifically , let M = G/K b e a n aturally redu ctiv e Riemann ian space with resp ect to G , g = k ⊕ m a natur al reduc- tiv e d ecomp osition, and assume that G acts almost effec tiv ely on M (i.e. the s ubgroup of elemen ts that act as the id en tit y transformation is discrete). Then th er e exists an an- alytic subgroup ¯ G of G and an analytic subgroup ¯ K of K so that M = ¯ G/ ¯ K and the metric is naturally reductive with resp ect to ¯ G and it arises from an Ad ∗ ( ¯ G )-in v arian t h function that is a nondegenerate quadratic p olynomial. The subgroup ¯ G is generated b y ¯ g = m + [ m , m ], w hic h is an id eal in g , and the Lie algebra of ¯ K is ¯ k = ¯ g ∩ k . The articles [42, 43] also co n tain the result that if h is a nondegenerate qu adratic Ad ∗ ( G )-in v arian t p olynomial on g ∗ and h ◦ f ∗ is p ositiv e d efinite, then th e metric defined by h is naturally reductiv e. Prop osition 3.15 L et M = G/K b e a R ie mannia n or pseudo-R ie mannian homo gene ous sp ac e. If a ∈ g is a r elative e quilibrium ve ctor at ( o, p ) , then λa is also a r elative e qui lib - rium ve ctor at ( o, λp ) for any λ ∈ R . Pro of. This result f ollo w s easily from lemma 3.3. ✷ 16 Prop osition 3.16 L et M = G/K b e a Riemannian or pseudo-R iemannian g.o. sp ac e. If ther e exists a ge o desic gr aph ξ so that ξ (0) = 0 and ξ is differ e ntiable at 0 , then ther e also exists a c orr esp onding ge o desic gr aph that is line ar. If, in addition, ξ is K -e quivariant, then the c orr esp onding line ar ge o desic gr aph is also K -e quivariant. Pro of. The differen tiabilit y of ξ and ξ (0) = 0 imp ly that ξ can b e w ritten as ξ = ξ (1) + ˜ ξ , where ξ (1) is linear and ˜ ξ has the prop ert y that lim λ → 0 ˜ ξ ( λp ) /λ = 0. ξ (1) is uniquely determined by ξ . It f ollo ws from prop osition 3.15 th at ξ λ ( p ) = ξ ( λp ) /λ is also a geod esic graph for any λ > 0. W e h a v e lim λ → 0 ξ λ ( p ) = ξ (1) ( p ), th u s ξ (1) is also a geod esic graph. If ξ is K -equiv ariant , then obvio usly ξ (1) is also K -equiv arian t. ✷ Theorem 3.17 L et M = G/K b e a Hamilto nian g.o. sp ac e. If K is c omp act, then ther e exists a K - e qu ivariant g e o desic gr aph. If K is c omp act and the sp ac e i s R ie mannia n, then ther e exists a K - e qu ivariant ge o desic gr aph ξ with the pr op erty that ξ ( λp ) = λξ ( p ) for al l λ ∈ R (i.e. ξ is first or der homo gene ous). Pro of. Due to th e co mpactness of K there exists a p ositiv e definite Ad ( K )-in v arian t scalar pro duct Q on g . F or any ( o, p ) ∈ T ∗ o M , consider the set of all relativ e equilibrium v ectors at ( o, p ), whic h is an affine subspace of g acco rding to p rop osition 3.4. Let the v alue of the geo desic graph at p b e that un iqu e elemen t of this affin e subspace whic h has the smallest norm with resp ect to Q . Sin ce Q is Ad ( K )-in v arian t, the geo desic graph defined in this wa y is ob viously K -equiv arian t. T aking in to consideration p r op ositio n 3.15, it is also ob vious that this geod esic graph has the prop ert y ξ ( λp ) = λξ ( p ) for all λ ∈ R if the Hamiltonian defines a Riemannian metric. ✷ It is easy to see that a sim ilar theorem w ith a similar pr oof holds for Lagrangian g.o. spaces as well. An Ad ( K )-inv arian t scalar pro duct on g exists also if M = G/K is a Riemannian g.o. space and the action of G on M is effectiv e (i.e . the compactness of K is not n ecessary . See e.g. [33], p rop osition 1 for a pr oof.) The pro of of p rop osition 3.16 also sho w s that if a geod esic graph ξ has the p rop erties that ξ ( λp ) = λξ ( p ) for any λ ∈ R and it is differentiable at 0, then ξ is linear. Consequently , we can state th e follo wing theorem: Theorem 3.18 L et M = G/K b e a R iemannian g.o. sp ac e and assume that the action of G on M is effe ctiv e . Then ther e exists at le ast one K -e q uivariant ge o desic gr aph ξ with the pr op erty tha t ξ ( λp ) = λξ ( p ) for any λ ∈ R . If ξ is differ e ntiable at 0 , then ξ is line ar and thus M is a nat ur al ly r e ductive sp ac e with r esp e ct to G . A geo desic graph th at hav e the stated prop erties can b e constructed in the same w ay as in the p ro of of 3.17. Th e main result in Szenthe’s pap er [2], whic h he obtained for affine g.o. manifolds with torsion-free affine co nnection and for co mpact K , is similar to theorem 3.18. O u r construction of the K -equiv arian t geo desic graph is simpler than that give n in [2] (constructions s im ilar to that in [2] can also b e fou n d in [6 , 9]). F or fur- ther results on the geodesic graphs of Riemannian g.o. spaces we refer the reader to [9, 18]. 17 In section 4 w e discu s s an example where h is a complicated function, nev ertheless h ◦ f ∗ is a homogeneous qu adratic p olynomial and it is also p ositiv e defi n ite, thus h still giv es rise to a Riemann ian metric on G/K . This metric has the g.o. prop ert y , b u t the geod esic graph, wh ic h is u nique in this example on an op en den s e set, is n ot linear and is not differen tiable at p = 0, and the metric is n ot n aturally reductiv e with resp ect to G , in accordance with theorems 3.17 and 3.18 . In addition to the nondifferentiabilit y at 0, the geodesic graph is also discontin uous along a one-dimensional su bspace (from wh ich 0 is excluded). As the example sho ws, in the Riemannian case the function h is not necessarily simple ev en though H | T o M ≡ H o = h ◦ f ∗ , and th us also h | m ∗ , where m ∗ is d efined as m ∗ = f ∗ ( T ∗ o M ), is a quadratic p olynomia l. Ho wev er, h | m ∗ is sufficien t for determinin g H o (since H o = h | m ∗ ◦ f ∗ ), and thus H . Therefore in order to sp ecify a Riemannian g.o . space it is sufficien t to s p ecify the p olynomial h | m ∗ , for whic h w e in tro du ce the notation h o = h | m ∗ . The g.o. prop ert y implies that there is an op en dense su bset N o of m ∗ suc h that at any p oin t b ∈ N o the deriv ativ e of h o has to b e zero in an y direction ad ∗ a ( b ), wh ere a ∈ g is suc h that ad ∗ a ( b ) ∈ m ∗ . That is to sa y , at an y p oin t b ∈ N o the equation ( dh o ( b ) | ad ∗ a ( b )) = 0 (42) has to hold f or all a ∈ g for whic h ad ∗ a ( b ) ∈ m ∗ . This equation can b e used in practice for finding suitable h o functions, i.e. for find ing g.o. m etrics or g.o. Hamiltonians, or to test whether a given metric or Hamiltonian function has the g.o. pr op erty . I n terms of H o , h o is giv en as h o = H o ◦ ( f ∗ ) − 1 , of course. The Ad ∗ ( K )-in v ariance of h o is n ecessa ry and sufficient for the G -in v ariance of the Hamiltonian function defi n ed by h o . If a ∈ k and b ∈ N o , th en ad ∗ a ( b ) ∈ m ∗ , thus (42 ) has to b e satisfied. Ho wev er, if h o is Ad ∗ ( K )-in v arian t, then (42) obviously holds if a ∈ k . The condition (42) is th erefore in teresting mainly for those elemen ts a of g whic h are not in k . The construction of g.o. Hamiltonian functions as H = h ◦ P can b e generalized in the follo wing wa y . Theorem 3.19 L et M b e a manifold and P a mapping T ∗ M → g ∗ , wher e g is a Lie algebr a of a Lie gr oup G , with the pr op erty [ X ( P | a ) , X ( P | b ) ] = X ( P | [ a,b ]) for al l a, b ∈ g . L et h b e a smo oth Ad ∗ ( G ) -invariant function. The Hamiltonian fu nction H = h ◦ P is G -invariant with r esp e ct to G i n the sense that H is c onstant alo ng the inte gr al curves of X ( P | a ) for al l a ∈ g . Any inte gr al curve of X H c oincides with an inte gr al curve of X ( P | a ) for some a ∈ g . In p articular, the inte gr al curve of X H starting at the p oint ( x, p ) ∈ T ∗ M c oincides with the inte gr al curve of X ( P | a ) , wher e a = dh ( P ( x, p )) , starting at ( x, p ) . In a more general f orm of the theorem the condition that h should b e smooth could b e relaxed. Certain notable dynamical systems, for example th e system of tw o p oin tlik e b o dies whic h in teract by the Newtonian gravita tional force (the K epler problem) and the harmonic oscillato r, admit a form u latio n in this framework with n oncomm utativ e groups G . Completely in tegrable systems can also b e f ormulated in the framew ork of theorem 3.19 with comm u tativ e symmetry groups. 18 4 Example In this section we discuss the example when G = S U (3) and K = S U (2) in order to give an illustration to the second part of section 3. Th e S U (3)-in v arian t metrics on S U (3) /S U (2), whic h is diffeomorphic to the sphere S 5 , constitute a t w o-parameter family . These metrics w ere describ ed e.g. in [44], where a co mplete description of the homogeneous metrics on the s pheres was give n. In [6 ] it w as found that all the S U (3)- in v ariant metrics on S U (3) /S U (2) h a v e the g.o. prop erty , but only a one-parameter su bfamily is naturally reductiv e with resp ect to S U (3). F urth er results, in p articular concernin g the geo desic graph, we re obtained in [9]. W e note that these metrics b elo ng to the t yp e of g.o. metrics whic h are naturally reductiv e with resp ect to a suitable larger s ymmetry group [9]. This larger group is U (3) in the p resen t case, and the stabilit y s u bgroup of th e origin is U (2). The Lie algebras of S U (3) and S U (2) are the follo wing: su (3) = g = k ⊕ m su (2) = k = span( A, B , C ) m = span( E 1 , E 2 , E 3 , E 4 , Z ) [ A, B ] = 2 C [ A, Z ] = 0 [ A, E 1 ] = − E 2 [ B , E 1 ] = E 3 [ C, E 1 ] = E 4 [ B , C ] = 2 A [ B , Z ] = 0 [ A, E 2 ] = E 1 [ B , E 2 ] = E 4 [ C, E 2 ] = − E 3 [ C, A ] = 2 B [ C, Z ] = 0 [ A, E 3 ] = E 4 [ B , E 3 ] = − E 1 [ C, E 3 ] = E 2 [ A, E 4 ] = − E 3 [ B , E 4 ] = − E 2 [ C, E 4 ] = − E 1 [ Z, E 1 ] = E 2 [ E 1 , E 2 ] = Z − 1 3 A [ E 2 , E 4 ] = 1 3 B [ Z, E 2 ] = − E 1 [ E 1 , E 3 ] = 1 3 B [ E 3 , E 4 ] = Z + 1 3 A [ Z, E 3 ] = E 4 [ E 1 , E 4 ] = 1 3 C [ Z, E 4 ] = − E 3 [ E 2 , E 3 ] = − 1 3 C. There exists one (up to m ultiplication by a constan t) q u adratic homogeneous inv arian t p olynomial on su (3): Y 1 = a ′ 2 + b ′ 2 + c ′ 2 + e 2 1 + e 2 2 + e 2 3 + e 2 4 + z 2 , (43) where a ′ , b ′ , c ′ , e 1 , e 2 , e 3 , e 4 , z den ote the co ordinates corresp ond ing to the basis v ectors A ′ = A √ 3 , B ′ = B √ 3 , C ′ = C √ 3 , E 1 , E 2 , E 3 , E 4 , Z of su (3). Y 1 defines a p ositiv e d efinite Ad -in v arian t qu ad r atic form on su (3), allo win g the iden tifi cation of su (3) and su (3) ∗ and implying the equiv alence of the coadjoin t and adjoin t actio ns of S U (3). The basis A ′ , B ′ , C ′ , E 1 , E 2 , E 3 , E 4 , Z is orthonormal with resp ect to the quad r atic form d efined b y Y 1 . W e use the same notation for the corresp ond ing orthonormal basis in su (3) ∗ . Y 1 can no w b e tak en as an inv ariant p olynomial on su (3) ∗ as well. f can b e used to id en tify T o M with m , and then the momen tu m mappin g restricted to T ∗ o M , i.e. f ∗ , is the trivial em b edd ing m → m ⊕ k . The p olynomial Y 1 comp osed with f ∗ th u s tak es th e form y 1 = Y 1 ◦ f ∗ = e 2 1 + e 2 2 + e 2 3 + e 2 4 + z 2 , (44) 19 where w e ha ve in tro duced the notatio n y 1 for Y 1 ◦ f ∗ . The metric on S U (3) / S U (2) corresp onding to y 1 is n aturally red uctiv e. In [6] it was found that the complete family of Riemannian g.o. metrics on S U (3) /S U (2) is giv en on T ∗ o M ≡ m by α ( e 2 1 + e 2 2 + e 2 3 + e 2 4 ) + β z 2 , α > 0 , β > 0 , (45) where α and β are real n umb ers. The metric (45) is naturally redu ctiv e if and only if α = β [6], which corresp onds to h = αY . The family of p olynomials (45) coincides w ith the complete family o f p ositiv e defin ite Ad ∗ ( K )-in v arian t quadratic homogeneous p olynomials on m . It is n ot difficult to verify that the metrics (45) also satisfy the condition (42). By solving the partial differen tial equations that express the Ad ∗ ( G )-in v ariance of a function w e find that the Ad ∗ ( G )-in v arian t f u nctions are of the form G ( Y 1 , Y 2 ), wh ere G is an arbitrary function of tw o v ariables an d Y 2 is the homogeneous third order p olynomial Y 2 = √ 3 σ 3 + z ( σ 2 − 2 σ 1 ) + 2 3 z 3 , (46) where σ 1 , σ 2 and σ 3 are th e follo wing Ad ∗ ( K )-in v arian t p olynomials: σ 1 = a ′ 2 + b ′ 2 + c ′ 2 (47) σ 2 = e 2 1 + e 2 2 + e 2 3 + e 2 4 (48) σ 3 = a ′ ( e 2 1 + e 2 2 − e 2 3 − e 2 4 ) + 2 b ′ ( e 1 e 4 − e 2 e 3 ) − 2 c ′ ( e 1 e 3 + e 2 e 4 ) . (49) W e hav e y 2 = Y 2 ◦ f ∗ = z ( e 2 1 + e 2 2 + e 2 3 + e 2 4 ) + 2 3 z 3 , (50) where the notation y 2 is in tro du ced f or Y 2 ◦ f ∗ . In order to get th e G function for whic h G ( Y 1 , Y 2 ) ◦ f ∗ equals (45) on e has to solv e the equations (44) and (50) for e 2 1 + e 2 2 + e 2 3 + e 2 4 and z . This in volv es the solution of a thir d order algebraic equation, therefore th e result is a complicated form ula that we do not write here. This example shows that the f u nction h (which is G ( Y 1 , Y 2 ) in the p resen t case) can b e complicated ev en though h ◦ f ∗ is a quadratic p olynomial. The ge o desic graph can b e calculated directly by solving the equations in lemma 3.3 or in lemma 2.4, as is done in [9] (it is the equatio n (15) that is actually used); it is not necessary for this to know h . Th e result, whic h can b e found written exp licit ly b elo w in equation (63) and in [9], has a relativ ely simple form. Th e geodesic graph can also b e calculated from the form u la ξ = dh ◦ f ∗ , where th e necessary deriv ativ es of h can b e determined fr om (42). As a third appr oac h, one can utilize the knowledge of the in v ariant p olynomials Y 1 and Y 2 to calculate dh ◦ f ∗ . Here we calculate the geod esic graph in this w ay , using (44), (50) and (45). W e hav e d ( G ( Y 1 , Y 2 )) = ∂ G ∂ Y 1 d Y 1 + ∂ G ∂ Y 2 d Y 2 , (51) th u s we hav e to calculate th e partial deriv ativ es of G . (45), (44) and (50) can b e written 20 as G ( y 1 , y 2 ) = αr 2 + β z 2 (52) y 1 = z 2 + r 2 (53) y 2 = 2 3 z 3 + z r 2 , (54) where r 2 = e 2 1 + e 2 2 + e 2 3 + e 2 4 . (55) W e hav e ∂ G ∂ y 1 = ∂ G ∂ r ∂ r ∂ y 1 + ∂ G ∂ z ∂ z ∂ y 1 (56) ∂ G ∂ y 2 = ∂ G ∂ r ∂ r ∂ y 2 + ∂ G ∂ z ∂ z ∂ y 2 . (57) F or ∂ G ∂ r and ∂ G ∂ z w e obtain ∂ G ∂ r = 2 αr ∂ G ∂ z = 2 β z (58) from (52). Th e partial deriv ativ es ∂ r ∂ y 1 , ∂ r ∂ y 2 , ∂ z ∂ y 1 and ∂ z ∂ y 2 can b e calculated b y taking partial deriv ativ es of the equations (53) and (54 ) with resp ect to y 1 and y 2 , and th en s olving the obtained four equations for ∂ r ∂ y 1 , ∂ r ∂ y 2 , ∂ z ∂ y 1 and ∂ z ∂ y 2 . Th e result is ∂ r ∂ y 1 = z 2 r 3 + 1 2 r ∂ r ∂ y 2 = z r 3 (59) ∂ z ∂ y 1 = − z r 2 ∂ z ∂ y 2 = − 1 r 2 . (60) T aking in to consideration (56) and (57 ) and using the resu lts (58), (59) and (60) w e obtain for ∂ G ∂ y 1 and ∂ G ∂ y 2 that ∂ G ∂ y 1 = α + ( α − β ) 2 z 2 r 2 (61) ∂ G ∂ y 2 = − ( α − β ) 2 z r 2 . (62) d Y 1 and d Y 2 are straigh tforw ard to calculate, and th e result for the geo desic graph is [ dG ( Y 1 , Y 2 ) ◦ f ∗ ]( e 1 E 1 + e 1 E 2 + e 3 E 3 + e 4 E 4 + z Z ) = 2 α ( e 1 E 1 + e 2 E 2 + e 3 E 3 + e 4 E 4 ) + 2 β z Z +( β − α ) 2 √ 3 z r 2 [( e 2 1 + e 2 2 − e 2 3 − e 2 4 ) A ′ +2( e 1 e 4 − e 2 e 3 ) B ′ − 2( e 1 e 3 + e 2 e 4 ) C ′ ] , (63) whic h agrees with the r esult obtained in [9], if we tak e int o consid eratio n the differences b et ween the defin itions in this pap er and in [9]. One difference that is worth n oting is 21 that in [9] the geo desic graph is d efi ned in such a wa y that only the k -comp onen t is k ept, i.e. the ob vious 2 α ( e 1 E 1 + e 2 E 2 + e 3 E 3 + e 4 E 4 ) + 2 β z Z part is subtracted. (63) is we ll defined on an op en dense s u bset of T ∗ o M , b ut it do es not ha ve we ll-defined v alues at r = 0 if α 6 = β . It can b e v erified using (27) and (28) that at z Z (i.e. when r = 0) all v ectors 2 β z Z + aA ′ + bB ′ + cC ′ , a, b, c ∈ R , are relativ e equilibriu m v ectors. The limit of (63) in the p oin ts charac terized by r = 0 and z 6 = 0 dep end s on the p ath (assumed to lie in the domain where r 6 = 0) along whic h th e limit is tak en, therefore the geo desic graph is necessarily discontin uous in these p oin ts. Sev eral other examples of Riemannian g.o . spaces ca n b e found in the literat ure (see e.g. [6, 9, 7]), w hic h w ould also b e int eresting to d iscu ss in a similar w ay . Ac kno wledgmen ts I would like to thank J´ anos Szenthe for prop osing this sub ject and for useful discussions, and L´ aszl´ o F eh´ er for his comments on the man uscript. I also thank the r eferees f or their constructiv e comments and for p ointing out the references [27, 44]. References [1] A. Ka plan, On t he ge ometry of gr oups of Heisenb er g typ e. B ull. London Math. 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