On bi-hamiltonian geometry of the Lagrange top

On bi-hamiltonian geometry of the Lagr ange top A V Tsigano v St.Petersbur g State University, St.Petersbur g, R ussia e–mail: tsiganov @mph.phys.spbu.ru Abstract W e consider three d ifferent incompatible bi- H amiltonian structures for th e Lagrange top, whic h h a ve the same foliation by symplectic lea ves. These b ivectors ma y b e asso ci- ated with the different 2-coboun daries in the Poisso n-Lichnero wicz cohomol ogy defined b y canonical biv ector on e ∗ (3). P A CS: 45.10 .Na, 45.40.Cc MSC: 70H20 ; 70H06; 37K10 1 In tro duction In recent years considerable pro gress has b een made in inv estigations o f the bi-integr able s ystems with functionally indep endent in tegrals of motion { H i , H j } = { H i , H j } ′ = 0 , i, j = 1 , . . . , n, where { ., . } and { ., . } ′ are compatible Poisson brack ets on the bi-hamiltonian manifold M . Historically the ma jority of them come from stationar y flows, restricted flows or the Lax equations of underly ing soliton systems (see r eferences in [1, 2, 8] ). Construction of integrals of motion f or suc h systems is usually based on the Lena rd-Magr i rec urrence rela tions. In or der to solve th e cor resp onding equations of motion in framework of the separation of v ariables metho d we have to use some suitable reductions of the Poisson biv ectors [2, 9]. The other class of bi-integrable systems come from r - matrix algebras, class ifications of 2- cob oundaries in the Poisson-Lichnero wicz cohomology and the separa tion of v aria bles metho d [4, 17, 18, 19]. The corresp onding Poisson brackets { ., . } and { ., . } ′ hav e a common foliation a s their symplectic leaf foliations. In this case we los e bene fits g iven b y bi-ha miltonian recurre nce relations, but we can obtain the se parated v ariables dir ectly . The main aim of this note is to discuss different bi-hamiltonian s tructures of b oth type s for the Lagr ange top. 2 The bi-hamiltonian manifolds In this section we descr ibe the ma nifolds where our bi-integrable s ystems will b e defined. Let M be a finite-dimensiona l Poisson manifold endow ed with a bivector P fulfilling the Jacobi condition [ P, P ] = 0 with res pe ct to the Schouten bra ck et [ ., . ]. W e will suppose that P has the constant cor ank k , dim M = 2 n + k , and that C 1 , . . . , C k are globa lly defined independent Casimir functions on M P dC a = 0 , a = 1 , . . . , k . The 2 n dimensional symplectic leaves of P form a symplectic foliation. 1 A bi-Hamiltonian manifold M is a smo oth (or complex) manifold endow ed with tw o com- patible bivectors P , P ′ such that [ P, P ] = [ P, P ′ ] = [ P ′ , P ′ ] = 0 . (2.1) Classification of compatible Poisson bivectors on low-dimensional Poisson manifolds is no wada ys a sub ject of intense resea rch. How ever the higher dimens ional problem is vir tually unt ouched. 2.1 In tegrals of motion from the Poisson biv ector s Let us cons ider bi-Ha miltonian manifold M with some kno wn bivectors P and P ′ . Moreov er, let as supp ose that there ar e k p olynomia l Casimir functions of the Poisson p encil P λ = P ′ − λP , H a ( λ ) = n a X i =0 H a i λ n a − i , H a 0 = C a , a = 1 , . . . , k , (2.2) such that n 1 + n 2 + · · · + n k = n and suc h that the differen tials of the co efficients H a i are linearly independent on M . The collection of the n bi-hamiltonia n vector fields X ( a ) i = P dH a i = P ′ dH a i − 1 , i = 1 , . . . , n a , k = 1 , . . . , a , (2.3) asso ciated with the Le nard-Mag ri sequences defined by the Casimirs H a ( λ ) is ca lled the Gel’fand– Zakhare vich system. The standard arg uments fr om the theory of Lenard–Mag ri chains sho w that all the co ef- ficient s H a i (2.2) pairwise commute with resp ect to b oth { ., . } and { ., . } ′ . It allows us to get nontrivial bi-integrable systems with integrals of motion H a i starting from the Casimir f unctions H a 0 = C a only . If there exists a folia tion of M , transversal to the sy mplectic leav es of P and compatible with the Poisson p encil (in a s uitable s ense), then the r estrictions of the Gel’fand–Zakha revich systems on sy mplectic leav es of P are separa ble in the so -called Darb oux-Nijenhuis v ariables [2]. Summing up, if we hav e tw o compatible Poisson bivectors P , P ′ and the Casimir functions C a of P , then we can get integrals of motion H i in the bi-inv olution using recur rence rela- tions (2.3) a nd, if we are luc ky , then we obtain the s eparated v ariables after s ome appro priate reduction. 2.2 The P oisson biv ectors from integrals of motion Let us consider bi-Ha miltonian manifold M with cano nical bivector P and some integrable system with integrals of mo tion H m . According to the Liouville-Arnold theo rem any integrable system admits separ ation of v ariables in the a ction-angle co o rdinates. According to [13] integrable s ystem on the symplectic leaves of M will b e said to b e separable in a set of ca nonical v aria bles ( p, q ) = ( p 1 , . . . , p n , q 1 , . . . , q n ) if there exist n separated equations of the form φ j ( p j , q j , α 1 , . . . , α n , C 1 , . . . , C k ) = 0 , det  ∂ φ i ∂ α j  6 = 0 , { q i , p i } = 1 . If we reso lve these eq uations with resp ect to para meters α 1 , . . . , α n one ge ts n independent int egra ls of mo tion α m = H m ( p, q , C ) , m = 1 , . . . , n, (2.4) as functions on the phase spa ce M with co o rdinates z = ( p, q , C ). These integrals of motion H i ( p, q , C ) ar e in the inv olution { H i , H j } f = 0 , i, j = 1 , . . . , n, (2.5) 2 with resp ect to the following bra ck et { ., . } f on M { q i , p j } f = δ ij f j ( p j , q j ) , (2.6) { p i , p j } f = { q i , q j } f = { p i , C j } f = { q i , C j } f = { C i , C j } f = 0 , which dep ends on n a rbitrar y functions f 1 ( p 1 , q 1 ) , . . . , f n ( p n , q n ) [16]. This bracket defines the Poisson bivector P f =   0 diag( f 1 , . . . , f n ) 0 − diag( f 1 , . . . , f n ) 0 0 0 0 0   , (2.7) compatible with the canonica l Poisso n bivector P o n M and s uch that P f dC a = 0 , a = 1 , . . . , k. (2.8) If all the f i 6 = 0, then P f has the same foliations by symplectic le av es a s P . So, it is e xplicit construction of the Poisson str uctures having the same foliation by symplectic leav es. Compatibility conditions (2.1), equatio ns (2.5) and (2.8) may b e chec k ed in any coor dinate system on M . So, for any integrable system o n M we can try to solve the following s ystem of equations [ P, P f ] = [ P f , P f ] = 0 , P f dC a = 0 , { H m , H l } f = 0 (2.9) with respec t to P f . O bviously enough, in their full generality equa tions (2.9) are too difficult to b e so lved b ecause it has infinitely many solutions la b eled by different s eparated co ordinates and their functions f = ( f 1 , . . . , f n ) [16]. In order to get particular s olution of the s ystem (2.9) we have to use some additio n a ssumptions or the couple of ans¨ atze. Nevertheless, using any known s olution P f of (2.9), we can easily to solve the following system of algebr aic equa tions P f dH m = n X l =1 F ml dH l , m = 1 ..n, (2.10) with resp ect to en tries of the n × n control matr ix F . According to [2] the eigenv alues of the control matrix F are the separ ated co o rdinates det( F − λ I) = n Y j =1 ( λ − q j ) . Solution o f the equations (2.9) – (2.10) may b e c onsidered a s a direct metho d of computation of the separ ated co o rdinates q j starting w ith given integrals of mo tion o nly . Remark 1 In fac t we p os tulate in (2.4) a nd (2.6) that o ur separa ted v ariables a re ” inv ariant” with resp ect to the Ca simirs, a s the one conside red in [1 5]. Remark 2 Bivectors P ′ fulfilling the compatibility co ndition [ P , P ′ ] = 0 are called 2 -co cycles in the Poisso n-Lichnero wicz cohomolog y defined by P on M [7]. The Lie deriv ativ e o f P along any vector field X on M P ′ = L X ( P ) (2.11) is 2-co b oundary , i.e. it is 2-co c ycle asso cia ted with the Liouville vector field X . F or such bivectors P ′ the compatibility conditions (2.1) are reduced to the single equa tion [ L X ( P 0 ) , L X ( P 0 )] = 0 , ⇔ [ L 2 X ( P 0 ) , P 0 ] = 0 (2.12) The seco nd Poisson-Lichnerowicz cohomolog y group H 2 P 0 ( M ) of M is pr ecisely the set of bivec- tors P 1 solving [ P 0 , P 1 ] = 0 modulo the solutions of the for m P 1 = L X ( P 0 ). W e can interpret H 2 P 0 ( M ) as the space of infinitesimal deformations o f the Poisson s tructure mo dulo trivial de- formations. F o r r egular Poisson manifolds co homology reflect the top olog y of the leaf space and the v a riation in the symplec tic str ucture as one pass es from one leaf to another . 3 In our case the co mpo nents of the Lio uville vector field X in the v ariables ( p, q , C ) ar e equal to X j =  F j ( q j , p j ) , j = 1 , . . . , n 0 , j = n + 1 . . . 2 n + k and bivector P f = L X ( P ) ha s the form (2.7) with f j ( q j , p j ) = − ∂ ∂ q j F j ( q j , p j ) . So, in fact in the separation o f v a riables metho d we are lo oking for sp ecial 2- cob oundaries (2.7) having the same folia tion by s ymplectic leav es a s canonical bivector P (2.8). Summing up, if w e have the canonical Poisson bivectors P , it’s Casimir functions C a and integrals o f motion H m for so me in tegra ble system, then we can try to g et co mpatible Poisson bivector P f from the equa tions (2.1), whic h immediately g ives rise to the corresp o nding separated v a riables. 3 The Lagrange top Let t wo vectors J = ( J 1 , J 2 , J 3 ) and x = ( x 1 , x 2 , x 3 ) are co or dinates on the Euclidean alg ebra e (3) ∗ with the Lie-Poisson bracket  J i , J j  = ε ij k J k ,  J i , x j  = ε ij k x k ,  x i , x j  = 0 , (3.1) where ε ij k is the totally skew-symmetric tensor. This bracket has tw o Casimir functions C 1 = | x | 2 ≡ 3 X k =1 x 2 k , C 2 = ( x, J ) ≡ 3 X k =1 x k J k . (3.2) Fixing their v alues o ne g ets a gener ic symplectic le af o f e (3) O ab : { x , J : C 1 = α 2 , C 2 = β } , which is a four-dimens ional symplectic ma nifold. As usua l we ident ify ( R 3 , ∧ ) and ( so (3) , [ ., . ]) by us ing the well known iso morphism o f the Lie algebra s z = ( z 1 , z 2 , z 3 ) → z M =  0 z 3 − z 2 − z 3 0 z 1 z 2 − z 1 0  , (3.3) where ∧ is the cro ss pro duct in R 3 and [ ., . ] is the matrix commutator in so (3). In these co ordinates the canonica l Poisso n bivector on e ∗ (3) is equal to P =  0 x M x M J M  . (3.4) The Lagr ange top is one of the most classic al exa mples of in tegrable systems with the following integrals of motion H 1 = J 3 , H 2 = J 2 1 + J 2 2 + J 2 3 + ax 3 , a ∈ R . (3.5) Here J and x denote resp ectively the angular momentum and the coo rdinates of the unit vector in the direction of g ravit y , a ll express ed in the b o dy frame. This is a sp ecial case o f ro tation of a r igid b o dy ar ound a fixed p oint in a ho mogeneous gravitational field, characterized by the following conditions: the rigid b o dy is rotationa lly symmetric, i.e. tw o of its three principal momen ts of iner tia co incide, and the fixed p oint lies on the axis of ro tational s ymmetry . The Lagr ange top is one o f the most clas sical ex amples o f integrable systems. The explicit formulae for th e positio n of t he bo dy in space were found b y Jaco bi [5]. F or an a ctual integration of the corr esp onding equations o f motion in terms o f elliptic functions see [6] a nd for a more mo dern account [3, 11]. 4 3.1 Recurrence relations According to [3] the inv ariant manifold o f the La grange top is isomo rphic to the in v ariant manifolds of o ne-gap so lutions o f the no n-linear Schr¨ odinger equa tion. Their bi-ha miltonian structures may b e identified a s well. So, for the Lagra nge top there a re tw o known Poisson bivectors P ′ compatible with the canonical bivector P (3.4) [3, 12]: P ′ 1 =    0 x 3 − x 2 0 0 0 − x 3 0 x 1 0 0 0 x 2 − x 1 0 0 0 0 0 0 0 0 − a 2 0 0 0 0 a 2 0 0 0 0 0 0 0 0    and P ′ 2 =     0 J 3 − J 2 0 a 2 0 − J 3 0 J 1 − a 2 0 0 J 2 − J 1 0 0 0 0 0 a 2 0 0 0 0 − a 2 0 0 0 0 0 0 0 0 0 0 0     . (3.6) They are 2 -cob oundarie s P ′ 1 , 2 = L X 1 , 2 ( P ) and the co rresp onding Liouville vector fields X 1 , 2 may b e obtained fro m the co rresp o nding vector fields from [19] b y using contraction o f so ∗ (4) to e ∗ (3). The Poisson p encil P λ = P ′ 1 − λP has one non-tr ivial p olynomial Casimir (2 .2) H 1 ( λ ) = C 1 , H 2 ( λ ) = 2 λ 2 C 2 + λH 2 + aH 1 , while the second Poisson p encil P λ = P ′ 2 − λP has tw o non-trivial Casimirs H 1 ( λ ) = − 2 λC 1 + H 2 , H 2 ( λ ) = − 2 λC 2 + aH 1 . Using the corre sp onding re currence relations 0 = P ′ 1 dC 1 , P ′ 1 H 1 = 0 , aP dH 1 = P ′ 1 dH 2 , P dH 2 = 2 P ′ 1 dC 2 , (3.7) and P ′ 2 dH 2 = 0 , P dH 2 = − 2 P ′ 2 dC 1 , P ′ 2 dH 1 = 0 , aP dH 1 = − 2 P ′ 2 dC 2 , (3.8) we can easily get integrals of motion H 1 , 2 (3.5) starting with the known Ca simir functions C 1 , 2 (3.2). The Poisson tensor s P ′ 1 and P ′ 2 are compatible to eac h other, i.e. [ P ′ 1 , P ′ 2 ] = 0. So, existence of the triad P , P ′ 1 , P ′ 2 of m utually compatible P oisson bivectors leads to tri-hamiltonian structure for the La grange top. Reducing ` a la Marsden– Ratiu this tri-Hamiltonian structure we ca n get the sepa rated v ariables for the La grange top. The r eduction may b e not unique, sinc e p os sibly different separ ated v ariables can b e cons tructed o n the symplectic leaf of P [9 ]. 3.2 P oisson struct ures havin g the same foliation by symplectic lea v es The g eneric s olution o f the equation P f dC 1 , 2 = 0 may b e parametrized by tw o vector functions f = ( f 1 , f 2 , f 3 ) and g = ( g 1 , g 2 , g 3 ) P f =  ( x, f ) x M f ⊗ ( x ∧ J ) + Q − [ f ⊗ ( x ∧ J ) + Q ] T − ( x ∧ J ) 3 g M  , (3.9) where Q =   x 2 ( x ∧ g ) 1 x 2 ( x ∧ g ) 2 x 2 ( x ∧ g ) 3 − x 1 ( x ∧ g ) 1 − x 1 ( x ∧ g ) 2 − x 1 ( x ∧ g ) 3 0 0 0   . Here ( u ⊗ v ) ij = u i v j and ( u ∧ v ) j is the j -th ent ry of the crosspr o duct u ∧ v of tw o vectors u and v . F o r any given integrable system on e ∗ (3) functions f and g hav e to satisfy o ne algebr aic equation { H 1 , H 2 } f = 0 (3.10) and ov erdetermined system o f alg ebro-differe nt ial equatio ns (2.1). T o solve these equations for the Lagra nge top we use some hypothesis abo ut the functions f . 5 3.2.1 So lution 1 If we put f 3 = 0 and ( x, f ) = 0, then one gets f 1 = − arcco s  x 3 | x |  ( x ∧ J ) 3 x 2 , f 2 = − arcco s  x 3 | x |  ( x ∧ J ) 3 x 1 and g 1 = arctan  x 1 x 2  ( x ∧ J ) 3 J 1 , g 2 = arctan  x 1 x 2  ( x ∧ J ) 3 J 2 , g 3 = − arcco s  x 3 | x |  ( x ∧ J ) 3 J 3 + x 3  arccos  x 3 | x |  − a rctan  x 1 x 2  ( x 1 J 1 + x 2 J 2 ) ( x 2 1 + x 2 2 )( x ∧ J ) 3 . The corr esp onding bivector (3 .9) we des ignate as P f 1 . In this c ase F = arctan “ x 1 x 2 ” 0 2 h arctan “ x 1 x 2 ” − arccos ( x 3 | x | ) i „ J 3 − x 3 ( x 1 J 2 + x 2 J 2 ) x 2 1 + x 2 2 « arccos ( x 3 | x | ) ! The eigenv alues of F are the separa ted v ariables q 1 = arctan  x 1 x 2  , q 2 = arccos  x 3 | x |  , which coincide with t he Euler angles φ and θ , resp ectively . The cano nically conjugated momenta read as p 1 = − J 3 , p 2 = − J 1 cos  arctan  x 1 x 2  + J 2 sin  arctan  x 1 x 2  . In v a riables ( q , p, C ) tw o compatible bivectors P and P f 1 hav e the standar d fo rm P =   0 I 0 − I 0 0 0 0 0   , P f 1 =   0 diag( q 1 , q 2 ) 0 − diag( q 1 , q 2 ) 0 0 0 0 0   . (3.11) Using v ariables ( q , p, C ) we can ea sily prov e that pro jections of the linear bivectors P ′ 1 and P ′ 2 (3.6) can not b e ass o ciated with the Euler a ngles. 3.2.2 So lution 2: If we put f 3 = 0 and ( x, f ) 6 = 0 then one gets f 1 = x 1 − i x 2 ( x ∧ J ) 3 J 2 + x 1 − i x 2 ( J 1 − i J 2 ) 2 | x | , f 2 = − x 1 − i x 2 ( x ∧ J ) 3 J 1 − x 1 − i x 2 ( J 1 − i J 2 ) 2 | x | and g m = − J 1 − i J 2 ( x ∧ J ) 3 J m . The corr esp onding bivector (3 .9) we des ignate as P f 2 . In this c ase F =    0 − i 2( x 2 + i x 3 ) a − J 2 + i J 3 x 2 + i x 3    6 and one gets complex sepa rated v ariables q 1 , 2 = − J 2 + i J 3 ± p ( J 2 + i J 3 ) 2 − 2 ia ( x 2 + i x 3 ) 2( x 2 + i x 3 ) , i 2 = − 1 . In v a riables ( q , p, C ) tw o compatible bivectors P and P f 2 hav e the form (2 .7) P =   0 I 0 − I 0 0 0 0 0   , P f 2 =   0 − diag( 1 q 1 , 1 q 2 ) 0 diag( 1 q 1 , 1 q 2 ) 0 0 0 0 0   , A t the first time these separa ted v ariables hav e been app ear in fr amework of the Sklyanin metho d and then hav e b een recov ered in [9] by reductio n of the the compatible linear Poisson bivectors P ′ 1 and P ′ 2 (3.6). In v ariables ( q , p, C ) these linear bivectors lo o k like P ′ 1 =     0 diag  a 2 q 1 , a 2 q 2  w 1 − diag  a 2 q 1 , a 2 q 2  0 w 2 − w 1 − w 2 0     and P ′ 2 =     0 − diag  a 2 4 q 2 1 , a 2 4 q 2 2  w 3 diag  a 2 4 q 2 1 , a 2 4 q 2 2  0 w 4 − w 3 − w 4 0     . The matrix elements of w k are brack ets { q i , C j } ′ m and { p i , C j } ′ m , which are so me nontrivial rational functions on the separ ated v a riables and Casimirs . F o r instance { q 1 , C 2 } ′ 1 = iq 1 p 2 q 1 − q 2 . W e ca n see t hat reductio ns of P ′ 1 and P f 2 on sy mplectic leaf o f P a re ident ical up to multiplication on a co nstant. Roug hly spe aking in this case reduction consists o f re moving the last rows and the last columns of P ′ 1 and P f 2 . 3.2.3 So lution 3: If we put f 1 = 0 then one g ets ( x, f ) = 1 and g m = ax m 2( x ∧ J ) 3 ( x 3 − i x 2 ) , f 2 = f 3 = 1 x 2 + i x 3 . (3.12) The corr esp onding bivector (3 .9) we des ignate as P f 3 . In this c ase F =   J 1 − i J 2 0 a ( x 1 − i x 2 ) + a ( x 1 − i x 2 ) 2 | x | ( J 1 − i J 2 ) 2 − x 1 − i x 2 | x | ( J 1 − i J 2 )   and the separated co o rdinates a re q 1 = J 1 − i J 2 , q 2 = − x 1 − i x 2 | x | ( J 1 − i J 2 ) . In v a riables ( q , p, C ) tw o compatible biv ectors P and P f 3 hav e the form (3.11) and, therefore, P f 3 is 2- cob oundary . The co rresp onding separa ted equatio ns, the Lax matrices, the r -matr ix formalism and the B¨ acklund tr ansformatio ns could b e found in [10]. The Poisson bivectors P f k , k = 1 , 2 , 3, ar e incompatible to e ach other, i.e. [ P f i , P f j ] 6 = 0 at i 6 = j . Moreov er, they are incompatible with the linear bivectors P ′ 1 , 2 (3.6) a s well. It means that we hav e differen t bi-hamiltonian structure s asso ciated with the Lagr ange to p. This fact deserves further inv estigation. 7 4 Another in tegrable sys tems on e ∗ (3) 4.1 The Gory ac hev-Chaplygin top The w ell-known Gor yac hev-Chaplygin case in rigid b o dy dynamics is describ ed by the following int egra ls of mo tion H 1 = J 2 1 + J 2 2 + 4 J 2 3 + ax 1 , H 2 = 2( J 2 1 + J 2 2 ) J 3 − ax 3 J 1 , a ∈ R . (4.1) On the fixed level ( x, J ) = 0 of the s econd Ca simir function the Hamilton function H 1 commutes with an additiona l cubic in tegral of motion H 2 . This fact ensur es the integrabilit y of the Goryac hev-Chaply gin ca se. Substituting anza ts (3 .9) in to (2.9) a t f 1 = 0 one gets the following s olution f 2 = 0 , f 3 = − 1 , and g 1 = − J 1 J 3 ( x ∧ J ) 3 , g 2 = − J 2 J 3 ( x ∧ J ) 3 , g 3 = J 2 1 + J 2 2 ( x ∧ J ) 3 (4.2) The corre sp onding p o lynomial P f (3.9) has b een obtained in [17] by using the r -matrix for- malism and the Sklyanin br ack ets. Remark 3 The s ame bivector P f could b e ea sier found by us ing Liouville vector field X with po lynomial entries: P f = L X ( P 0 ) , X = X X m ( z ) ∂ /∂ z m , z = ( x, J ) . If we supp ose that X m ( z ) are arbitr ary quadratic po lynomials on M X m = n X ij c ij m z i z j , m = 2 , . . . , 6 , c m ij ∈ C , (4.3) then from (2.9) one eas ily gets X =  0 , x 3 J 2 , − x 2 J 2 , − J 1 J 3 , 0 , − J 2 2 − J 2 3  . (4.4) In contrast with r ational functions (4.2) here we hav e simple p oly nomials only . As a use full by-product w e directly pro ve that th e biv ector P f is 2- cob oundary in the corresp onding Poisson- Lichnero wicz co homology . In this case the co ntrol ma trix F is eq ual to F =  2 J 3 − 1 − J 2 1 − J 2 2 0  , (4.5) and its eigenv alues q 1 , 2 = J 3 ± q J 2 1 + J 2 2 + J 2 3 (4.6) satisfy to the following dynamica l eq uations ( − 1) j ( q 1 − q 2 ) ˙ q j = 2 q P ( q j ) 2 − | x | 2 a 2 q 2 j , P ( λ ) = λ 3 − λH 1 + H 2 . (4.7) These equa tions a re reduced to the Ab el-J acobi equa tions a nd, therefore , they ar e so lved in quadratures . 8 4.2 The Sok olo v system on the sphere Let us c onsider another in tegrable at ( x, J ) = 0 system on e ∗ (3) [14] with integrals of motion second and fourth orde r: H 1 = J 2 1 + J 2 2 + 2 J 2 3 + a ( x 3 J 1 − J 3 x 1 ) + 2 bJ 3 , H 2 = ( J 2 1 + J 2 2 + J 2 3 )(2 J 3 + 2 b − ax 1 ) 2 . (4.8) Using the s ame anzats (4.3) for the comp o nent s X m ( z ) of the Liouville vector field as for the Goryac hev-Chaply gin top one g ets the same solution (4.4) of the equations (2.9). In this case the control matrix reads a s F =   J 3 1 2(2 J 3 + 2 b − ax 1 ) 2( J 2 1 + J 2 2 + J 2 3 )(2 J 3 + 2 b − ax 1 ) J 3   . (4.9) Its eige n v alues coincide with the Chaplygin v ariables (4.6 ), which are the s eparated v ariables for the Sokolov system to o . 4.3 The Ko wa levski top. Let us consider the Kowalevski to p with the following int egra ls of mo tion H 1 = J 2 1 + J 2 2 + 2 J 2 3 − 2 bx 1 , (4.10) H 2 =  ( J 1 + i J 2 ) 2 + 2 b ( x 1 + i x 2 )  ( J 1 − i J 2 ) 2 + 2 b ( x 1 − i x 2 )  . Solution of the equations (2.9) has b een co nstructed in [18] by using the r - matrix forma lism and the reflection equation alg ebra. In our no tations this solution is defined by f 1 = − 2 J 1 − (2 x 1 J 2 − x 2 J 1 )  b ( x 2 J 2 + x 3 J 3 ) + J 1 J 2 3  J 2 2 ( x ∧ J ) 3 , f 2 = J 2 − J 2 1 + b x 1 J 2 − x 1 (2 J 2 3 − b x 1 ) ( x ∧ J ) 3 + x 2 J 3 (2 J 1 J 3 + b x 3 ) J 2 ( x ∧ J ) 3 + x 1 ( J 1 J 3 + b x 3 ) 2 J 2 2 ( x ∧ J ) 3 f 3 = J 3 − bx 2 2 J 3 J 2 ( x ∧ J ) 3 + ( J 1 J 3 + b x 3 )( J 1 ( x ∧ J ) 3 − b x 1 x 2 ) J 2 2 ( x ∧ J ) 3 and g 1 = b ( x 1 J 1 + x 3 J 3 ) ( x ∧ J ) 3 + bx 2 ( J 2 1 − J 2 3 + b x 1 ) J 2 ( x ∧ J ) 3 + J 1 ( J 1 J 3 + b x 3 ) 2 J 2 2 ( x ∧ J ) 3 , g 2 = 2 bx 2 J 1 ( x ∧ J ) 3 + b 2 x 2 2 + J 1 J 3 ( J 1 J 3 + b x 3 ) J 2 ( x ∧ J ) 3 + bx 2 J 3 ( J 1 J 3 + b x 3 ) J 2 2 ( x ∧ J ) 3 , g 3 = bx 3 J 1 ( x ∧ J ) 3 + bx 2 ( J 1 J 3 + b x 3 ) J 2 ( x ∧ J ) 3 + J 3 ( J 1 J 3 + b x 3 ) 2 J 2 2 ( x ∧ J ) 3 . The co rresp onding separated v ariables q 1 , 2 are the famous Kow alevski v ariables [18]. In these v ar iables bivectors P and P f hav e the form (3.11). It allo ws us to prov e that the second biv ector P f is the 2-cob oundary in the cor resp onding Poisson-Lichnerowicz cohomolo gy . 5 Concluding remarks The ma in r esult in this pap er is constr uction of the different bi-Hamiltonian structur es for the Lagra nge top, which have the s ame foliation by symplectic leaves. The co rresp onding three 9 incompatible Poisson bivectors may be asso cia ted with the 2-cob oundaries in the Poisson- Lichnero wicz co homology defined b y c anonical bivector P on e ∗ (3). As a la st r emark, we obser ve that similar bi-hamilto nian str uctures e xist for so me other int egra ble sys tems on e ∗ (3), for instance fo r the K ow alevski top. The simila r 2- cob oundaries in the Poisson-Lichnerowicz cohomology o n so ∗ (4) were conside red in [1 9]. The resea rch was par tially supp orted by the RFBR gra nt 0 6-01 -0014 0. References [1] M. 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