New Variables of Separation for the Steklov-Lyapunov System

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 8 (2012), 012, 14 pages New V ariables of Separation for the Steklo v–Ly apuno v System A ndr ey V. TSIGANO V St. Petersbur g State University, St. Petersbur g, R ussia E-mail: andr ey.tsiganov@gmail.c om Receiv ed Octob er 31, 2011, in f inal form March 12, 2012; Published online March 20, 2012 h ttp://dx.doi.org/10.3842/SIGMA.2012.012 Abstract. A rigid b o dy in an ideal f luid is an imp ortant example of Hamiltonian systems on a dual to the semidirect product Lie algebra e (3) = so (3) n R 3 . W e presen t the bi- Hamiltonian structure and the corresp onding v ariables of separation on this phase space for the Steklov–Ly apuno v system and it’s gyrostatic deformation. Key wor ds: bi-Hamiltonian geometry; v ariables of separation 2010 Mathematics Subje ct Classific ation: 70H20; 70H06; 37K10 1 In tro duction Rigid b o dy dynamics in an ideal incompressible f luid is rich with problems interesting from mathematical p oint of view, in particular, the research of in tegrable problems. Certainly , the most famous are the three in tegrable cases under the names of Kirc hhof f, Clebsch, Steklov and Ly apunov. The latter t wo cases are more in teresting b ecause there are no a obvious symmetry groups asso ciated with the additional integrals of motion. All these classical cases were discov ered and carefully studied in the 18th and 19th cen- turies [ 5 ]. F or instance, the Kirchhof f equations for the Clebsc h and Steklov–Ly apunov cases w ere f irst solv ed explicitly by K¨ otter after some m ysterious separation of v ariables [ 11 , 12 ]. A t the momen t no separation whic h is alternativ e to his original separation of v ariables is kno wn for these systems, even though there is a lot of literature dedicated to the problem, including the theta-functions solutions, asso ciated with the Lax matrices, and the detailed geometric de- scriptions of the inv arian t surfaces on which the motions ev olve, see b o oks [ 1 , 2 ] and references within. In this pap er w e apply the bi-Hamiltonian geometry to the direct calculation of v ariables of new separation for the Steklo v–Lyapuno v system and it’s Rubano vsky generalization. W e hav e to notice right aw ay , that our main purp ose is the dev elopment of the bi-Hamiltonian geometry instead of integration of particular equations of motion using separation of v ariables metho d. An integrable system is separable, if there are n separation relations Φ i ( u i , p u i , H 1 , . . . , H n ) = 0 , i = 1 , . . . , n, with det  ∂ Φ i ∂ H j  6 = 0 , (1.1) connecting single pairs ( u i , p u i ) of canonical v ariables of separation with the n functionally indep enden t Hamiltonians H 1 , . . . , H n . Solving these relations in terms of p u i one gets the Jacobi equations and the corresp onding additively separable complete integral of the Hamilton–Jacobi equation W = n X i =1 Z u i p u i ( u 0 i , α 1 , . . . , α n ) du 0 i , α j = H j . 2 A.V. Tsiganov Of course, any v ariables of separation are determined up to the trivial transformation u i → ˜ u i = f i ( u i , p u i ) , (1.2) whic h preserve all the prop erties of algebraic curves def ined by separated relations ( 1.1 ). Ho we- v er, if w e hav e another separated relations for the same in tegrals of motion Ψ i ( v i , p v i , H 1 , . . . , H n ) = 0 , i = 1 , . . . , n, with det  ∂ Ψ i ∂ H j  6 = 0 , whic h can not b e reduced to initial ones ( 1.1 ) b y trivial change of v ariables ( 1.2 ), we usually sa y ab out dif feren t v ariables of separation. Of course, any t wo families of canonical v ariables on a given phase space are related by a generic canonical transformations v i = g i ( u 1 , . . . , u n , p 1 , . . . , p n ) , p v i = h i ( u 1 , . . . , u n , p 1 , . . . , p n ) . (1.3) In contrast with ( 1.2 ) the notion of generic transformations ( 1.3 ) allows us to study relations b et w een distinct algebraic curv es, for instance co vering of the algebraic curv es (see works of P oincar´ e, Hum b ert, F rey , Kani, Kuhn, Shask a) or curves with isogenous Ab el v arietes (see w orks of Richelot, Bro ck, Ha yashida, Nishi, Ibukiy ama, Katsura, v an W amelen). Such relations giv e us a lot of examples of the reductions of Ab elian in tegrals (see works of Hermite, Goursat, Burkhardt, Brioschi, Bolza) and, therefore, they may b e a source of new ideas in n um b er theo- ry , algebraic geometry and mo dern cryptograph y (see works of T ate, F altings, Zarhin, Lange, McMullen, Merel). Nev ertheless, motiv ation for the search of suc h dif feren t v ariables of separation do not pure mathematical, b ecause dif ferent v ariables of separation may b e useful in dif ferent p erturbation theories [ 18 , 22 , 24 ], as well as in distinct procedures of quantization and v arious metho ds of qualitativ e analysis, etc. The milestones of the v ariables separation technique include the works of St¨ ac k el, Levi- Civita, Eisenhart, Benenti and others. The ma jority of results was obtained for a very sp ecial class of in tegrable systems, important from the physical p oin t of view, namely for the natural Hamiltonian systems with quadratic in momen ta integrals of motion on cotangen t bundles to Riemannian manifolds. The Ko walevski and Chaplygin results on separation of v ariables for the systems with higher-order in tegrals of motion hav e b een missed out of this sc heme un til recen tly [ 18 , 22 , 24 ]. In the Steklov–Ly apunov case we ha ve quadratic integrals of motions, but the phase space is the P oisson manifold instead of the cotangen t bundle to Riemannian manifold. So, in this case we can use neither the Levi-Civita criteria, nor the Eisenhart–Benenti theory . Belo w w e sho w how v ariables of separation for the giv en integrable system may b e calculated without any additional information (Killing tensors, Lax matrices, r -matrices, links with soliton equations etc.). 2 Steklo v–Ly apuno v system F ollowing Kirchhof f, w e consider the p otential motion of a f inite rigid b o dy submerged in an inf initely large volume of irrotational, incompressible, inviscid f luid that is at rest at inf inity , so that the induced motion of particles of the f luid is completely determined by the motion of the b o dy [ 9 ]. In this case, the motion of rigid b o dy is describ ed by the classical Kirc hhof f equations ˙ M = M × Ω + p × U, ˙ p = p × Ω , (2.1) here x × y stands for the vector pro duct of three-dimensional vectors. V ectors M and p are the impulsiv e momen tum and the impulsive force while Ω and U are the angular and linear New V ariables of Separation for the Steklo v–Lyapuno v System 3 v elo cities of the bo dy . All these vectors in R 3 are expressed in the b o dy frame attac hed to the b o dy originating at the cen ter of buo yancy [ 9 ]. A rigid b o dy in the ideal f luid is an imp ortant example of Hamiltonian systems on a dual to Lie algebra e (3) = so (3) n R 3 . The dual space e ∗ (3) is the Poisson manifold endow ed with the canonical Lie–Poisson brack ets  M i , M j  = ε ij k M k ,  M i , p j  = ε ij k p k ,  p i , p j  = 0 , (2.2) where ε ij k is a totally sk ew-symmetric tensor. There are t wo Casimir elemen ts C 1 = h p, p i = | p | 2 ≡ 3 X i =1 p 2 i , C 2 = h p, M i ≡ 3 X i =1 p i M i , (2.3) where h x, y i means scalar pro duct of t wo three-dimensional v ectors x, y ∈ R 3 . As usual [ 9 ], elemen t M ∈ so (3) is identif ied with three-dimensional vector M ∈ R 3 using w ell known isomorphism of the Lie algebras ( R 3 , × ) and so (3), [ · , · ] z = ( z 1 , z 2 , z 3 ) → z µ =   0 z 3 − z 2 − z 3 0 z 1 z 2 − z 1 0   , (2.4) where × is a cross product, [ · , · ] is a matrix comm utator and index µ means a 3 × 3 an tisymmetric matrix asso ciated with the vector z . Using this agreemen t w e can rewrite canonical Poisson biv ector on e ∗ (3) in the follo wing compact form P =         0 0 0 0 p 3 − p 2 0 0 0 − p 3 0 p 1 0 0 0 p 2 − p 1 0 0 p 3 − p 2 0 M 3 − M 2 − p 3 0 p 1 − M 3 0 M 1 p 2 − p 1 0 M 1 − M 1 0         =  0 p µ p µ M µ  . (2.5) The Hamilton function H = H ( p, M ) and Lie–Poisson brack ets ( 2.2 ) allow us to def ine the Hamiltonian equations of motion ˙ M = M × ∂ H ∂ M + p × ∂ H ∂ p , ˙ p = p × ∂ H ∂ M . (2.6) These generic Euler’s equations on e ∗ (3) coincide with the Kirchhof f equations ( 2.1 ), if H ( p, M ) is a second-order p olynomial in v ariables M and p . Remark 1. The Lie–P oisson dynamics on e ∗ (3) can b e interpreted as resulting from reduction b y the symmetry group E (3) of the full dynamics on the t welv e-dimensional phase space T ∗ E (3). Here, symmetry means that the Hamiltonian that describes the dynamics in T ∗ E (3) is an in v ariant to actions of E (3), i.e., one can translate the inertial frame or rotate it in any direction without af fecting the equations of motion [ 15 ]. The Steklo v–Lyapuno v case of the rigid b o dy motion is c haracterized b y the follo wing second- order homogeneous p olynomial integrals of motion H 1 = h M , M i − 2 h A p, M i − h ( A 2 + 2 A ∨ ) p, p i , H 2 = h A M , M i + 2 h A ∨ p, M i −  h A 3 p, p i − tr A 2 h A p, p i  , (2.7) 4 A.V. Tsiganov where wedge denotes an adjoint matrix, i.e. a cofactor matrix A ∨ = (det A ) A − 1 . F or integra- bilit y A has to b e symmetric matrix, which may b e reduced to the diagonal form A = diag( a 1 , a 2 , a 3 ) , a i ∈ R , (2.8) using linear canonical transformations of e ∗ (3). F rom ph ysical p oin t of view it means that the b o dy axes can alwa ys be chosen so that A is diagonal. Remark 2. In [ 17 ] Steklov found integrable Hamiltonian H 2 , whereas Lyapuno v prov ed inte- grabilit y of the Kirc hhof f equations with Hamiltonian H 1 in [ 14 ]. The generic family of the Steklo v–Lyapuno v in tegrable systems w as studied b y Kolosov in [ 10 ]. 2.1 Separation of v ariables by K¨ otter The explicit integration of the classical Steklo v–Lyapuno v systems via separation of v ariables had b een f irst made by F. K¨ otter in 1900 [ 11 ]. Here we w ant to add some new details in the kno wn coincidence of the K¨ otter v ariables of separation v 1 , 2 with the elliptic co ordinates on the sphere. According to [ 23 , 25 ] there is a Poisson map, which iden tif ies the Steklov–Ly apunov system with a system that describ e s motion on the surface of a unit tw o-dimensional sphere S 2 in a fourth-degree p olynomial p otential f ield. This dynamical system is separable in standard elliptic coordinates on the sphere, and the in verse Poisson map allows us to get complete solution of the Steklov–Ly apuno v system. Prop osition 1. If B = tr A − A and C = √ B 2 − 4 A ∨ ar e diagonal matric es with entries B =   a 2 + a 3 0 0 0 a 1 + a 3 0 0 0 a 1 + a 2   , C =   a 2 − a 3 0 0 0 a 3 − a 1 0 0 0 a 1 − a 2   , then the Poisson map f : ( p, M ) → ( x, J ) , define d by x = ( M − B p ) × p | ( M − B p ) × p | , J = M + C  x, x × p  + , (2.9) wher e [ y , z ] + i = n =3 X j,k =1 | ε ij k | y j z k , r elates manifold e ∗ (3) with c o or dinates ( p, M ) and c otangent bund le T ∗ S 2 to the unit two- dimensional spher e S 2 with c o or dinates ( x, J ) . The pro of consists in the verif ication of the Lie–P oisson brack ets betw een v ariables x and J  J i , J j  = ε ij k J k ,  J i , x j  = ε ij k x k ,  x i , x j  = 0 , and calculation of the corresponding Casimir functions h x, x i = | x | 2 = 1 , and h x, J i = 0 . So, this coa join t orbit of e ∗ (3) with co ordinates x and J is simplectomorphic to cotangent bundle T ∗ S 2 to the unit t wo-dimensional sphere S 2 , see for instance [ 15 ]. New V ariables of Separation for the Steklo v–Lyapuno v System 5 In verse Poisson map f − 1 : ( x, J ) → ( p, M ) looks like p = αJ + β ( x × J ) , M = J − C  x, x × p  + , (2.10) where functions α , β on x , J are solutions of the following equations C 1 = h p, p i = α 2 | J | 2 + β 2 | x × J | 2 , C 2 = h p, M i = α 2  | J | 2 h x, A x i − h ( x × J ) , A ( x × J ) i  + 2 αβ h J , A ( x × J ) i + β 2  h ( x × J ) , B ( x × J ) i − 2 h J, A J i  + α | J | 2 . Prop osition 2. The Poisson map ( 2.9 ) r elates the Steklov inte gr al of motion H 1 ( p, M ) ( 2.7 ) with the natur al Hamilton function on T ∗ S 2 H 1 ( x, J ) = h J, J i + 4 h x, B x i  (tr A C 1 − C 2 ) − C 1 h x, B x i  + 4 C 1 h x, A ∨ x i + 2 tr A C 2 − tr( A 2 + 4 A ∨ ) C 1 . (2.11) The Lyapunov inte gr al H 2 ( p, M ) ( 2.7 ) is e qual to H 2 ( x, J ) = h J, A J i − 4 h x, A ∨ x i  C 1 h x, B x i − (tr A C 1 − C 2 )  + 2 tr A C 2 − 2 det A C 1 . (2.12) Her e C 1 , 2 ar e values of the Casimir functions ( 2.3 ) . The pro of of this prop osition and all the details ma y b e found in [ 23 , 25 ]. Remark 3. It is w ell-known that the generic lev el sets of the Casimir functions (coadjoin t orbits) on e ∗ (3) are only dif feomorphic to the cotangen t bundle T ∗ S 2 . It allows us to directly connect the Kirc hhof f equations with the equations of motion b y geo desic on S 2 , how ever, at C 2 6 = 0 w e hav e to destro y the standard symplectic structure on T ∗ S 2 b y adding some “monop ole” terms [ 15 ]. In the Steklo v–Lyapuno v case w e use the P oisson map ( 2.9 ), whic h preserv e s the standard symplectic structure on T ∗ S 2 . As a punishment for this preserv ation of the standard symplectic structure we hav e to consider p otential motion on S 2 instead of the geo desic motion. The Hamiltonians H 1 , 2 ( x, J ) ( 2.11 ), ( 2.12 ) on T ∗ S 2 are separable in the elliptic (sphero coni- cal) co ordinates v 1 , 2 , which are zero es of the function e ( λ ) = ( λ − v 1 )( λ − v 2 ) det( λ − A ) = h x, ( λ − A ) − 1 x i ≡ 3 X i =1 x 2 i λ − a i . (2.13) This v ariables of separation satisfy the follo wing separated relations Φ( v i , p v i ) = det( v i − A ) p 2 v i + h ` ( v i ) , ` ( v i ) i = 0 , i = 1 , 2 , (2.14) where the three-dimensional v ector ` ( λ ) is the so-called K¨ otter vector with en tries ` i ( λ ) = √ λ − a i 2  M i +  2 λ + a i − tr A  p i  dep ending on the auxiliary v ariable λ (spectral parameter). The explicit solution of the cor- resp onding Ab el–Jacobi equations in theta-functions was given in [ 11 ]. In order get initial v ariables p , M as functions on time v ariable w e ha ve to substitute solutions of the Ab el–Jacobi equations into the v ariables on T ∗ S 2 x i = s ( v 1 − a i )( v 2 − a i ) ( a j − a i )( a k − a i ) , J i = 2 ε ij k x j x k ( a j − a k ) v 1 − v 2  ( a i − v 1 ) p v 1 − ( a i − v 2 ) p v 2  , where ( i, j , k ) is p erm utation of (1 , 2 , 3), and then in to the v ariables p and M ( 2.10 ). Another mo dern v erif ication of the K¨ otters calculations ma y b e found in [ 4 , 8 ]. 6 A.V. Tsiganov 3 Calculation of the v ariables of separation in bi-Hamiltonian geometry W e can only guess how K¨ otter in ven ted the v ariables of separation v 1 , 2 , whic h coincide with the elliptic coordinates on the auxiliary t wo-dimensional sphere, because he ga ve no explanations of calculations in v ery brief comm unication [ 11 ]. It is clear that behind the striking formulas there must b e a certain geometric idea, but the domain of applicabilit y of this idea is usually restricted by a partial mo del under consideration. F or instance, w e can not apply the K¨ otter separation to the Kow alevski top and Ko walevski separation to the Steklov–Ly apuno v system etc. Our aim is to discuss some algorithm of calculation of the v ariables of separation in the framew ork of the bi-Hamiltonian geome try , which is applicable to many kno wn integrable sys- tems [ 7 , 18 , 20 , 21 , 22 , 24 , 26 ]. In fact this algorithm consists of the follo wing steps: • calculate the second Poisson brack et compatible with canonical one starting with the given in tegrals of motion in the in volution with resp ect to this canonical Poisson brac ket; • if the Poisson brack ets ha v e dif ferent symplectic lea ves, calculate a pro jection of the second brac ket on symplectic lea ves of the f irst brac ket; • calculate co ordinates of separation as eigenv alues of the corresp onding recursion op erator; • calculate the canonically conjugated momen ta with resp ect to the f irst Poisson brack et; • calculate the separated relations. The input of algorithm is a set of integrals of motion and canonical P oisson brac ket, whereas output is a set of separated relations. All details ab out construction of a suitable pro jection are discussed in [ 7 ]. Because v ariables of separation ( 1.1 ) are def ined up to canonical transformations u i → f ( u i , p u i ) on the f irst step of this algorithm we hav e to narrow the search space using some artif icial tricks. It is a main tec hnical problem of this method. The second technical problem is the calculation of the momen ta conjugated to obtained co ordinates, see [ 7 , 18 , 22 , 24 ]. 3.1 P olynomial and rational P ois son brac k ets on e ∗ (3) Bi-Hamiltonian structures can b e seen as a dual formulation of in tegrability and separability , in the sense that they substitute a hierarch y of compatible Poisson structures to the hierarch y of functions in in volution, which may b e treated either as in tegrals of motion or as v ariables of separation. So, our f irst step is calculation of the second Poisson bivector P 0 compatible with kinematic Poisson bivector P . According to [ 20 , 26 ] any separable system is a bi-in tegrable system, i.e. in tegrals of mo- tion H k ( 1.1 ) are in bi-in volution { H i , H k } = { H i , H k } 0 = 0 v , i, k = 1 , . . . , n, (3.1) with resp ect to compatible P oisson brac kets {· , ·} and {· , ·} 0 asso ciated with the P oisson bivec- tors P and P 0 , so that [ [ P, P ] ] = 0 , [ [ P , P 0 ] ] = 0 , [ [ P 0 , P 0 ] ] = 0 . (3.2) Here [ [ · , · ] ] is the Sc houten brac k et. The def inition of the second brack et {· , ·} 0 in term of v ariables of separation may b e found in [ 20 , 26 ]. F or the given integrable system f ixed by a kinematic bivector P and a tuple of in tegrals of motion H 1 , . . . , H n bi-Hamiltonian construction of v ariables of separation consists in a direct New V ariables of Separation for the Steklo v–Lyapuno v System 7 solution of the equations ( 3.1 ) and ( 3.2 ) with respect to an unkno wn bivector P 0 . The main problem is that the geometrically inv arian t equations ( 3.1 ), ( 3.2 ) ha ve a’priory inf inite num b er of solutions [ 18 , 20 , 21 , 22 , 24 , 26 ]. In order to get a searc h algorithm of ef fectively computable solutions we hav e to narro w the searc h space b y using some non-in v ariant additional assumptions. According to [ 20 , 21 ] hereafter w e assume that P 0 has the same foliations b y symplectic lea ves as P , i.e. that P 0 dC 1 , 2 = 0 (3.3) and P 0 do esn’t ha ve any other Casimir elements. The geometric meaning of this restriction is discussed in [ 20 , 21 ]. In fact it allows us to av oid calculations of the pro jection of the second brac ket on the symplectic leav es of the f irst brack et. In the Steklo v–Lyapuno v case solving equations ( 3.1 ), ( 3.2 ) and ( 3.3 ) in the space of ho- mogeneous second-order p olynomial biv ectors and of rational bivectors with the second-order homogeneous numerators and linear denominators we obtain the follo wing tw o prop ositions. Prop osition 3. If c and d ar e two numeric thr e e-dimensional ve ctors, so that h c, c i = 0 , then e quations ( 3.2 ) and ( 3.3 ) on e ∗ (3) have a p olynomial solution P 0 1 =      h c, p i p µ h c, M i p µ + ( p × M ) ⊗ c + 1 2  1 α + α h c, d i   p ⊗ p − h p, p i  + α ( c × p ) ⊗ ( d × p ) ∗ h c, M i M µ + h d, p i p µ + 1 2  1 α + α h c, d i  ( p × M ) µ − α  ( c × p ) × ( d × M )  µ      , (3.4) α ∈ C , and a r ational solution P 0 2 = 1 h c, p i P 0 1 + 1 h c, p i  h c, M i + α (1 + α 2 h c, d i ) 2 α 2 h c, d i h c × p, d i  P , (3.5) c omp atible to e ach other, i.e. [ [ P 0 1 , P 0 2 ] ] = 0 . As ab ov e × is a cross product, the an tisymmetric matrix z µ is def ined by v ector z ( 2.4 ) and the matrix ( x ⊗ y ) ij = x i y j is determined by a pair of v ectors x and y . In order to explain this notations we write out the corresp onding Poisson brac kets { p i , p j } 0 1 = ε ij k h c, p i p k , { p i , M j } 0 1 = ε ij k h c, M i p k + ( p × M ) i c j + 1 2 α  1 + α 2 h c, d i  p i p j − 3 X l =1 p 2 l ! + α ( c × p ) i ( d × p ) j , (3.6) { M i , M j } 0 1 = ε ij k  h c, M i M k + h d, p i p k  + ε ij k  1 2 α  1 + α 2 h c, d i  ( p × M ) k − α  ( c × p ) × ( d × M )  k  . The sec ond brack ets are equal to { p i , p j } 0 2 = ε ij k p k , { p i , M j } 0 2 = { p i , M j } 0 1 h c, p i + ε ij k p k h c, p i  h c, M i + α (1 + α 2 h c, d i ) 2 α 2 h c, d i h c × p, d i  , (3.7) { M i , M j } 0 2 = { M i , M j } 0 1 h c, p i + ε ij k M k h c, p i  h c, M i + α (1 + α 2 h c, d i ) 2 α 2 h c, d i h c × p, d i  . W e hav e to stress that this brack ets are def ined o ver complex f ield b ecause h c, c i = c 2 1 + c 2 2 + c 2 3 = 0. 8 A.V. Tsiganov Substituting the Poisson brack ets {· , ·} 0 1 , 2 ( 3.6 ), ( 3.7 ) into ( 3.1 ) and solving the resulting equations in the space of the second-order homogeneous p olynomials H 1 , 2 one gets the follo wing prop osition. Prop osition 4. The Steklov–Lyapunov inte gr als of motion H 1 , 2 ( p, M ) ( 2.7 ) satisfy the e qua- tion ( 3.1 ) at α = 1 and c = − 1 p ( a 1 − a 2 )( a 2 − a 3 )( a 3 − a 1 )  √ a 2 − a 3 , √ a 3 − a 1 , √ a 1 − a 2  , d = p ( a 1 − a 2 )( a 2 − a 3 )( a 3 − a 1 )  1 √ a 2 − a 3 , 1 √ a 3 − a 1 , 1 √ a 1 − a 2  . (3.8) The pro of is a straightforw ard calculation. Remark 4. In fact, p olynomial biv ector P 0 1 has b een obtained in [ 21 ] as an inciden tal re- sult b y in vestigation of the Poisson biv ectors on the Lie algebra so ∗ (4) and the corresp onding in tegrable cases in the Euler equations on so ∗ (4). No w w e reco ver this bivector by solving equations ( 3.1 ), ( 3.2 ) and ( 3.3 ) for the Steklo v–Lyapuno v system. Let us brief ly discuss the bi-Hamiltonian structure related with the K¨ otter v ariables of sepa- ration. Elliptic co ordinates on the sphere S 2 ( 2.13 ) are associated with the p olynomial Poisson biv ector P 0 ell = L X P , where L X is a Lie deriv ative along the v ector f ield X = P X j ∂ j with the following entries: X i = 0 , X i +3 =  x × A ( x × J )  i , i = 1 , 2 , 3 . Biv ector P 0 e is compatible with P and has the same foliation b y symplectic lea ves as P . Using the Poisson map ( 2.9 ), ( 2.10 ) w e can easily express P 0 ell in the initial v ariables p , M . It will b e a rational bivector P 0 ell = R/Q , where R is a bivector with fourth-order homoge- neous polynomial entries and Q = | ( M − B p ) × p | 2 is a fourth-order p olynomial as w ell. So, w e could directly calculate the K¨ otter v ariables solving equations ( 3.1 ), ( 3.2 ) and ( 3.3 ) in the corresp onding space of rational biv ectors. 3.2 Calculation of v ariables of separation The bi-inv olutivit y of the integrals of motion ( 3.1 ) is equiv alent to the existence of con trol matrix F def ined by P 0 dH = P  F dH  , or P 0 dH i = P n X j =1 F ij dH j , i = 1 , . . . , n. The additional assumption ( 3.3 ) ensures that F is a non-degenerate matrix and the eigenv alues of F are the desired v ariables of separation [ 20 , 21 ]. Moreo ver, for the so-called St¨ ac kel sepa- rable systems the suitable normalized left eigenv ectors of the control matrix F form the St¨ ac kel matrix S [ 20 , 21 , 22 , 24 ]. In this case separated relations ( 1.1 ) are af f ine equations in integrals of motion H k . Let us calculate the con trol matrices for the Steklo v–Lyapuno v system and, for brevity , in tro duce three constan ts τ k = tr A k ≡ 3 X i =1 a k i , k = 0 , 1 , 2 , New V ariables of Separation for the Steklo v–Lyapuno v System 9 and some linear functions on v ariables p , M ρ k = h c, A k p i , σ k = h c, A k M i , k = 0 , 1 , 2 , (3.9) whic h are related to each other via the Casimir functions ( 2.3 ) on e ∗ (3). F or instance, C 1 = 3 X i =1 p 2 i = − 1 2  τ 2 1 − τ 2  ρ 2 0 + 2( τ 1 ρ 1 − ρ 2 ) ρ 0 − ρ 2 1 . (3.10) In this notations control matrices asso ciated with the Hamiltonians H 1 , 2 ( 2.7 ) and the Poisson biv ectors P 0 1 , 2 ( 3.4 ), ( 3.5 ) lo ok like F 1 = − 2 ρ 1 2 ρ 0 σ 1 + ρ 2 − τ 1 ρ 1 − σ 0 − ρ 1 + τ 1 ρ 0 ! , F 2 =     σ 0 − ρ 1 ρ 0 − τ 1 3 2 σ 1 + ρ 2 − τ 1 ρ 1 ρ 0 2 τ 1 3     . (3.11) No w w e can simply calculate the desired v ariables of separation using tw o control matrices. Namely , let u 1 , 2 b e eigen v alues of the control matrix F 2 B ( λ ) = det( F 2 − λ ) = ( λ − u 1 )( λ − u 2 ) = λ 2 −  σ 0 − ρ 1 ρ 0 + τ 1 3  λ − 2 τ 2 1 9 + 2(2 ρ 1 + σ 0 ) τ 1 3 ρ 0 − 2( σ 1 + ρ 2 ) ρ 0 , (3.12) whereas the eigen v alues of F 1 b e doubled momenta 2 p u 1 , 2 , so that the characteristic p olynomial has the form A ( λ ) = det( F 1 − λ ) = ( λ − 2 p u 1 )( λ − 2 p u 2 ) = λ 2 + ( σ 0 + 3 ρ 1 − τ 1 ρ 0 ) λ + 2 ρ 1 ( σ 0 + ρ 1 ) − 2 ρ 0 ( σ 1 + ρ 2 ) . (3.13) Another equiv alent def inition of momen ta p u i is given by a relation p u i = h c, p i 2 u i − h c, M i 2 + h c, d × p i 3 , i = 1 , 2 . No w we can pro ve the follo wing Prop osition 5. On symple ctic le aves of e ∗ (3) variables u 1 , 2 and p u 1 , 2 ar e c anonic al variables { u i , p u i } = 1 , { u i , p u i } 0 1 = 2 p u i , { u i , p u i } 0 2 = u i , i = 1 , 2 . with r esp e ct to c anonic al Poisson br acket ( 2.2 ) . The pro of consists of the calculation of the Poisson brac kets betw een co ef f icients of c harac- teristic p olynomials A ( λ ) ( 3.13 ) and B ( λ ) ( 3.12 ). No w we ha ve to determine an in verse transformation from v ariables u 1 , 2 and p u 1 , 2 to initial v ariables p , M . Firstly , using the def initions ( 3.13 ), ( 3.12 ) and the relation ( 3.10 ), we express f ive linear functions ( 3.9 ) via v ariables of separation and Casimir functions ρ 0 = 2 p u 1 − p u 2 u 1 − u 2 , ρ 1 = 2 τ 1 − 3 u 1 3( u 1 − u 2 ) p u 1 − 2 τ 1 − 3 u 2 3( u 1 − u 2 ) p u 2 , ρ 2 = ( u 1 − u 2 ) C 1 4( p u 1 − p u 2 ) −  τ 2 + τ 2 1 9  p u 1 − p u 2 2( u 1 − u 2 ) + 2 τ 1 3 p u 1 u 2 − p u 2 u 1 u 1 − u 2 + ( p u 1 u 2 − p u 2 u 1 ) 2 4( p u 1 − p u 2 )( u 1 − u 2 ) , σ 0 = u 1 + 2 u 2 u 1 − u 2 p u 1 − 2 u 1 + u 2 u 1 − u 2 p u 2 , (3.14) σ 1 = − ρ 2 + τ 1 ρ 1 3 − 2( p u 1 u 1 − p u 2 u 2 ) τ 1 3( u 1 − u 2 ) + ( p u 1 − p u 2 ) u 1 u 2 u 1 − u 2 . It is easy to see that ρ k and σ k are symme tric functions in u 1 , 2 , p u 1 , 2 . 10 A.V. Tsiganov Secondly , we determine the initial v ariables p , M as functions on ρ k and σ k : p i = c i  a j a k ρ 0 − ( a j + a k ) ρ 1 + ρ 2  , i = 1 , 2 , 3 , ( i, j, k ) = (1 , 2 , 3) , M i = c i ρ 0  C 2 − σ 1 ( a i ρ 0 − ρ 1 ) + σ 0  ρ 2 − τ 1 ρ 1 + a i ( a j + a k ) ρ 0  . (3.15) Here c i are entries of the vector c ( 3.8 ), C 1 , 2 are the Casimir functions on e ∗ (3) ( 2.3 ) and ( i, j, k ) means the p ermutation of (1 , 2 , 3). Matrices F 1 , 2 ( 3.11 ) in canonical v ariables of separation lo ok like F 1 = S  2 p 1 0 0 2 p 2  S − 1 , F 2 = S  q 1 0 0 q 2  S − 1 , where the St¨ ac kel matrix S is equal to S =   6 2 τ 1 − 3 u 1 6 2 τ 1 − 3 u 2 1 1   . (3.16) The notion of the St¨ ac kel matrix S allo ws us to easily get the separated relations ( 1.1 ) and prov e that canonical v ariables u , p u are the v ariables of separation for the Steklov–Ly apunov system. Prop osition 6. In the Steklov–Lyapunov c ase the c anonic al variables u 1 , 2 ( 3.12 ) and p u 1 , 2 ( 3.13 ) satisfy the fol lowing sep ar ate d r elations Φ( u, p u ) =  u 2 − τ 1 3  H 1 + H 2 + ϕ 3 ( u ) p 2 u + φ 3 ( u ) = 0 , u = u 1 , 2 , p u = p u 1 , 2 , (3.17) wher e cubic p olynomials ϕ 3 ( u ) and φ 3 ( u ) ar e e qual to ϕ 3 ( u ) =  u 3 2 + u 3  τ 2 1 − 3 τ 2  − 4 27 ( τ 1 − 3 a 1 )( τ 1 − 3 a 2 )( τ 1 − 3 a 3 )  , φ 3 ( u ) = C 1 u 3 2 + C 2 u 2 −  C 2 τ 1 + C 1 τ 2 1 3 − C 1 τ 2 2  u + C 2  7 τ 2 1 9 − τ 2  + C 1  τ 3 1 27 + 2 τ 1 τ 2 3 − 2 τ 3 3  . (3.18) The pro of consists of substituting in tegrals of motion H 1 , 2 ( 2.7 ) in terms of v ariables of separation ( 3.15 ) into the separated relations ( 3.17 ). So, in the Steklov–Ly apunov case equations of motion are linearized on Jacobian of the gen us t wo hyperelliptic curve def ined by the equation Φ( u, p u ) = 0 ( 3.17 ) and the system of the Ab el–Jacobi equations has the standard form Z u 1 ∞ du p ( u ) ϕ 3 ( u ) + Z u 2 ∞ du p ( u ) ϕ 3 ( u ) = β 1 t + γ 1 , Z u 1 ∞ udu p ( u ) ϕ 3 ( u ) + Z u 2 ∞ udu p ( u ) ϕ 3 ( u ) = β 2 t + γ 2 . (3.19) Here p ( u ) means the function p u on u obtained from the equation ( 3.17 ), β 1 , 2 are certain con- stan ts dep ending only on the c hoice of the Hamiltonian ( H 1 or H 2 ) and γ 1 , 2 are tw o constan ts. Solving these equations with resp ect to u 1 , 2 ( t, γ 1 , 2 ) and substituting these solutions in to the ex- pressions ( 3.14 ) and ( 3.15 ) we f inally get the initial v ariables p , M as functions on time v ariable t and six constants H 1 , 2 , C 1 , 2 and γ 1 , 2 . New V ariables of Separation for the Steklo v–Lyapuno v System 11 Remark 5. In order to giv e an explicit theta-functions solution, one can apply the standard mac hinery of the W eierstrass ro ot functions describing in v ersion of the hyperelliptic quadra- tures ( 3.19 ), completely similar to solution of Jacobi’s geodesic problem or Neumann’s particular case of the Clebsc h system [ 5 , 8 , 27 ]. Using shift of co ordinates u i → 2 3 ( a 1 + a 2 + a 3 − 3 u i ) w e can rewrite separated relations ( 3.17 ) in the following form p 2 u = C 1 u 3 + ( C 1 tr A − C 2 ) u 2 + 4 ˜ H 1 u + 4 ˜ H 2 det( u − A ) , (3.20) where ˜ H 1 = − H 1 − C 1 tr  A 2 + 4 A ∨  + 2 C 2 tr A , ˜ H 2 = H 2 + 2 C 1 det A − 2 C 2 tr A ∨ . Separated relations ( 3.20 ) ha ve the same form as the K¨ otter separated relations ( 2.14 ). Ho wev er, w e ha ve to point out that if p and M are real v ariable, then u 1 , 2 are complex functions in contrast with the real K¨ otter v ariables v 1 , 2 . In terms of u 1 , 2 and p u 1 , 2 symmetric functions on the K¨ otter v ariables v 1 , 2 lo ok lik e v 1 + v 2 = P ( u 1 , u 2 , p u 1 , p u 2 ) Q ( u 1 , u 2 , p u 1 , p u 2 ) , v 1 v 2 = R ( u 1 , u 2 , p u 1 , p u 2 ) T ( u 1 , u 2 , p u 1 , p u 2 ) . Here P , Q , R , T are the sixth-order p olynomials in momen ta, Q , T are the sixth-order p olyno- mials in co ordinates, whereas P and R are the sev enth- and eigh th-order p olynomials in co or- dinates. W e did not f ind a foreseeable expressions for these polynomials or their com binations. In any case v ariables u , p u and v , p v are related by non-trivial canonical transformation ( 1.3 ). Remark 6. The Steklo v–Lyapuno v system on e ∗ (3) coincides with the Steklov system on so ∗ (4) after some linear change of phase v ariables [ 3 ]. It is a twisted P oisson map, whic h p ermutes f irst and second Lie–Poisson brack ets on e ∗ (3) and so ∗ (4) [ 19 ]. W e supp ose that v ariables u 1 , 2 and p u 1 , 2 coincide with the complex v ariables of separation for the Steklo v system on so (4) in tro duced in [ 6 , 13 ] up to this c hange of v ariables and transformation of the canonical momenta asso ciated with p ermutation of the P oisson brac kets. W e thank one of the referees for the reference on these pap ers. Algebro-geometric relations of this complex co ordinates u 1 , 2 with the real K¨ otter co ordinates v 1 , 2 is discussed in [ 8 ]. 3.3 The Rubano vsky system Let us consider a non trivial in tegrable generalization of the Steklo v–Lyapuno v system discov ered b y Rubanovsky [ 16 ] ˆ H 1 = H 1 + 2 h b, p i , ˆ H 2 = H 2 + h b, (tr A − A ) p i − h b, M i , (3.21) where H 1 , 2 are giv en b y ( 2.7 ) and b = ( b 1 , b 2 , b 3 ) is a constan t v ector. This deformation describes the motion of a gyrostat in an ideal f luid under the action of the Archimedes torque, whic h arises when the barycen ter of the gyrostat do es not coincide with its volume center. The problem of separation of v ariables for the Rubanovsky systems w as unsolv ed up un til now. The Rubano vsky integrals of motion are non-homogeneous second-order p olynomials and, therefore, it is natural to solve the equations ( 3.1 ) and ( 3.2 ) in the space of non-homogeneous second-order p olynomial bivectors and in the similar space of rational biv ectors. 12 A.V. Tsiganov Prop osition 7. Inte gr als of motion ˆ H 1 , 2 ( p, M ) ( 3.21 ) ar e in bi-involution ( 3.1 ) with r esp e ct to the Poisson br ackets asso ciate d with the p olynomial Poisson bive ctor ˆ P 0 1 = P 0 1 + h b, c i         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p 3 − p 2 0 0 0 − p 3 0 p 1 0 0 0 p 2 − p 1 0         (3.22) and the r ational Poisson bive ctor ˆ P 0 2 = 1 h c, p i ˆ P 0 1 + 1 h c, p i  h c, M i + α (1 + α 2 h c, d i ) 2 α 2 h c, d i h c × p, d i  P , (3.23) wher e P 0 1 is the Poisson bive ctor for the Steklov–Lyapunov system ( 3.4 ) in which α = 1 and ve ctors c and d ar e given by ( 3.8 ) . The pro of is a straightforw ard v erif ication of the equations ( 3.1 ) and ( 3.2 ). Remark 7. If c = ( c 1 , c 2 , p − c 2 1 − c 2 2 ) is an arbitrary v ector, d = ( c 1 α − 2 , 0 , 0) and α → ∞ , then the P oisson bivectors ˆ P 0 1 , 2 yield bi-Hamiltonian structures on e ∗ (3) asso ciated with the Lagrange top [ 20 ]. F ollowing the same line as previously , let us calculate the con trol matrices asso ciated with the Hamiltonians ˆ H 1 , 2 ( 3.21 ) and the P oisson bivectors ˆ P 0 1 , 2 ( 3.22 ), ( 3.23 ) ˆ F 1 = F 1 − h b, c i 2  0 0 1 0  , ˆ F 2 = F 2 − h b, c i 2 ρ 0  0 0 1 0  . Eigen v alues of these matrices are new canonical v ariables of separation ˆ u 1 , 2 and ˆ p u 1 , 2 , so that ˆ F 1 = S  2 ˆ p 1 0 0 2 ˆ p 2  S − 1 , ˆ F 2 = S  ˆ u 1 0 0 ˆ u 2  S − 1 , where S is the same St¨ ack el matrix ( 3.16 ). These v ariables are simply related with the previous one ˆ u 1 , 2 = u 1 + u 2 2 ± 1 2 s ( u 1 − u 2 ) 2 − 4 h b, c i ρ 0 , ˆ p u 1 , 2 = p u 1 + p u 2 2 ± 1 2 p ( p u 1 − p u 2 ) 2 − h b, c i ρ 0 . Initial v ariables p , M are the same functions on ˆ ρ k and ˆ σ k p i = c i  a j a k ˆ ρ 0 − ( a j + a k ) ˆ ρ 1 + ˆ ρ 2  , i = 1 , 2 , 3 , ( i, j, k ) = (1 , 2 , 3) , M i = c i ˆ ρ 0  C 2 − ˆ σ 1 ( a i ˆ ρ 0 − ˆ ρ 1 ) + ˆ σ 0  ˆ ρ 2 − τ 1 ˆ ρ 1 + a i ( a j + a k ) ˆ ρ 0  , where four functions ˆ ρ k ( ˆ u, ˆ p u ) = ρ k ( u, p ) and ˆ σ 0 ( ˆ u, ˆ p u ) = σ 0 ( ˆ u, ˆ p u ) is given by ( 3.14 ): ˆ ρ 0 = 2 ˆ p u 1 − ˆ p u 2 ˆ u 1 − ˆ u 2 , ˆ ρ 1 = 2 τ 1 − 3 ˆ u 1 3( ˆ u 1 − ˆ u 2 ) ˆ p u 1 − 2 τ 1 − 3 ˆ u 2 3( ˆ u 1 − ˆ u 2 ) ˆ p u 2 , ˆ σ 0 = ˆ u 1 + 2 ˆ u 2 ˆ u 1 − ˆ u 2 ˆ p u 1 − 2 ˆ u 1 + ˆ u 2 ˆ u 1 − ˆ u 2 ˆ p u 2 , ˆ ρ 2 = ( ˆ u 1 − ˆ u 2 ) C 1 4( ˆ p u 1 − ˆ p u 2 ) −  τ 2 + τ 2 1 9  ˆ p u 1 − ˆ p u 2 2( ˆ u 1 − ˆ u 2 ) + 2 τ 1 3 ˆ p u 1 ˆ u 2 − ˆ p u 2 ˆ u 1 ˆ u 1 − ˆ u 2 + ( ˆ p u 1 ˆ u 2 − ˆ p u 2 ˆ u 1 ) 2 4( ˆ p u 1 − ˆ p u 2 )( ˆ u 1 − ˆ u 2 ) New V ariables of Separation for the Steklo v–Lyapuno v System 13 and one function is dif ferent ˆ σ 1 = h b, c i 2 − ˆ ρ 2 + τ 1 ˆ ρ 1 3 − 2( ˆ p u 1 ˆ u 1 − ˆ p u 2 ˆ u 2 ) τ 1 3( ˆ u 1 − ˆ u 2 ) + ( ˆ p u 1 − ˆ p u 2 ) ˆ u 1 ˆ u 2 ˆ u 1 − ˆ u 2 . This shift of ˆ σ 1 acts only on M v ariables ( 3.15 ). Prop osition 8. In the Rub anovski c ase the sep ar ate d r elations have the fol lowing form ˆ Φ( ˆ u, ˆ p u ) =  ˆ u 2 − τ 1 3  ˆ H 1 + ˆ H 2 + ϕ 3 ( ˆ u ) ˆ p 2 u −  h b, c i 2 ˆ u 2 + h b, D c i 3 ˆ u + 2 h b, D ∨ c i 9  ˆ p u (3.24) + φ 3 ( ˆ u ) − h b, c i 2  ˆ u 4 + τ 1 2  + h b, c ih b, A c i 2 = 0 , ˆ u = ˆ u 1 , 2 , ˆ p u = ˆ p u 1 , 2 . Her e cubic p olynomials ϕ 3 and φ 3 ar e given by ( 3.18 ) and D = tr A − 3 A . The proof consists of substituting in tegrals of motion ˆ H 1 , 2 in terms of v ariables of separa- tion ˆ u , ˆ p u in to the separated relations ( 3.24 ). As ab ov e, w e can prov e that the equations of motion are linearized on the Jacobian v ariety of the gen us t wo hyperelliptic curv e def ined b y ( 3.24 ). 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