On natural Poisson bivectors on the sphere

On natural P oisson biv ectors on the sphere A V Tsigano v St.Petersbur g State Univers ity, St.Peter sbur g, Russia e–mail: andr ey.tsiganov@gmail.c om Abstract W e discuss the concept of natural P oisson bivectors , which allo ws us to consider the o ver- whelming ma jorit y of kn o wn integrable systems on the sph ere in framework of bi-Hamiltonian geometry . 1 In tro duction The Hamilton-Jacobi theory seems to b e o ne of the most powerful metho ds of inv estiga tion the dy- namics o f mechanical (holono mic and nonholonomic) and c ontrol systems. Besides its fundamental asp ects such as its relatio n to the action integral and gene r ating functions of symplectic maps, the theory is known to be very use ful in in tegrating the Hamilton equatio ns using the v ariables separa - tion tec hnique. The milestones of this techn ique include the works o f St¨ ack el, Levi-Civita, Eisenhart, W o o dhouse, Kalnins, Miller, Benenti and others. The ma jority o f r esults was obtained for a very sp ecial clas s of integrable systems, imp orta nt from the physical p oint of view, namely for the systems with quadratic in momenta int egrals of motio n. The Kow alev s ki, Chaplygin and Go ryac hev results on separatio n of v a r iables for the systems with hig her order integrals of mo tion missed out of this scheme. Bi-Hamiltonian s tructures c a n b e seen as a dual formulation of integrability and separability , in the sense that they substitute a hiera rch y of compatible Poisson str uc tur es to the hier arch y of functions in inv olution, whic h may b e treated either a s in tegrals of motion or a s v a riables of separ ation for some dynamical system. The Eisenhart-Be nent i theo ry was embedded into the bi-Hamiltonian set-up using the lifting o f the confor mal Killing tensor that lies at the heart of Benenti’s co nstruction [8, 1 5]. The concept o f natur al Poisson bivectors allows us to gener alize this constr uction and to study s ystems with quadr atic and higher order in teg rals of motion in framework of a single theory [31]. The aim o f this note is to br ing together all the known examples of na tural Poisson biv ectors o n the sphere, b ecause a go o d example is the bes t sermon. Some of these Poisson bivectors hav e b een obtained and presented ea r lier in differ e nt co or dinate sy stems a nd notations. Here we prop os e the unified des cription of this known and few new bivectors using so-ca lle d geo desic Π and p otential Λ matrices [31]. In some sense we prop ose new form for the old c onten t and b elieve that this unifica tion is a first step to the g e ometric analys is o f v ario us natural systems on the sphere, whic h reveals what they hav e in common and indicates the most suitable strateg y to obtain and to analyze their solutions. The cor resp onding in tegrable natural systems on tw o-dimensio nal unit sphere S 2 are rela ted to rigid bo dy dyna mics. In orde r to describ e these sys tems w e will use the angula r momentum vector J = ( J 1 , J 2 , J 3 ) and the Poisson vector x = ( x 1 , x 2 , x 3 ) in a moving frame of coor dinates attached to the principa l axes of inertia [4 ]. The Poisson brack ets b etw ee n these v ariables  J i , J j  = ε ij k J k ,  J i , x j  = ε ij k x k ,  x i , x j  = 0 , (1.1) may be a sso ciated to the Lie-Poisson algebra of the three- dimensional Euclidean a lgebra e (3) with t wo Casimir elements C 1 = | x | 2 ≡ 3 X k =1 x 2 k , C 2 = h x, J i ≡ 3 X k =1 x k J k . (1.2) Below w e alwa ys put C 2 = 0. 1 As usual a ll the results ar e present ed up to the linear canonical transfor ma tions, whic h consist of rotations x → α U x , J → U J , where α is a n arbitra ry parameter and U is an or thogonal cons tant matrix, and shifts x → x , J → J + S x , where S is an arbitra ry 3 × 3 sk ew-symmetric constant matrix [4 , 1 6]. If the s quare in tegra l o f motio n C 2 = ( x, J ) is eq ual to zero , rigid b o dy dynamics may b e restricted on the unit sphere S 2 and we can use standard spherica l co or dinate system o n it’s cotangent bundle T ∗ S 2 x 1 = sin φ sin θ, x 2 = cos φ sin θ, x 3 = cos θ , J 1 = sin φ cos θ sin θ p φ − cos φ p θ , J 2 = cos φ cos θ sin θ p φ + sin φ p θ , J 3 = − p φ . (1.3) W e use these v ariables in order to determine and classify the natural Poisson biv ectors on T ∗ S 2 up to the p oint canonica l transformatio ns. As far as the org anization of this pap er is c o ncerned, in Section 2 w e briefly in tro duce the notions of bi-Hamiltonian geometry relev ant for subsequent sectio ns . In particular, we discuss the concept of natural P oisson biv ec to rs on cotang e nt bundles to Riemannian manifolds , which allows us to genera lize classical Eisenhart-Benenti theory . In Section 3 we discuss the bi-Hamiltonia n c la ssification of bi- int egrable systems on the sphere. Sectio n 4 is devoted to the separ able natural systems coming from auxiliary bi-Hamilto nia n systems. 2 Some issues in the geometry of bi-Hamiltonian manifolds A bi-Hamiltonian manifold M is a smooth manifold endo wed with a pair of compatible Poisson biv ectors P and P ′ such tha t [ P, P ′ ] = 0 , [ P ′ , P ′ ] = 0 , (2.1) where [ ., . ] is the Schouten brack et. This means that every linear c ombination of P a nd P ′ is still a Poisson biv ector. If P is in vertible Poisson bivector on M , one can intro duce the so-c alled Nijenhuis o p e r ator (or hereditary , or recursio n) N = P ′ P − 1 . (2.2) If N has, at every p o int, th e max imal num b er o f different functionally indep endent eigenv alues u 1 , . . . , u n , then M is said to be a re g ular bi-Hamiltonian manifold. 2.1 Bi-in tegrable syst ems Let us co nsider a family o f bi-integrable systems for which there are functiona lly independent integrals of motio n H 1 , . . . , H n in the bi- involution { H i , H j } = { H i , H j } ′ = 0 , i, j = 1 , . . . , n, (2.3) with res p e c t to a pair o f compatible Poisson br ack ets { ., . } and { ., . } ′ defined by P a nd P ′ . Ther e are three known distinct constructions of bi-integrable systems, see [3 1] . Firstly , if M is a r egular bi- Ha miltonian manifo ld endow e d with inv er tible Poisson bivector P , then we can construct recur sion op erato r N (2.2) and, as us ua l, functions H k = 1 2 k tr N k (2.4) form a bi-Hamiltonian hier arch y o n M , i.e. the Lenard re la tions hold P ′ d H k = P d H k +1 , for all k ≥ 1 . Using these relations we can g et all the integrals of motion starting with the Hamilton function H 1 . 2 Remark 1 The natural obstacle for existence of the bi-Hamiltonian systems is dis c ussed in [5]. F or- tunately , we can use these rare bi-Ha miltonian systems (natur al or no n-natural) a s auxiliary sys tems for the construction o f an infinite family of non bi-Hamiltonian separ able systems. Namely , s e cond sp ecial but mor e fundamen tal construction of in tegrable systems was originally formulated by Ja cobi when he inv en ted elliptic co o rdinates and successfully applied them to s olve several imp ortant mechanical problems: ”The main difficulty in int e gr ating a given differ ent ial e quation lies in intr o ducing c onvenient variables, which ther e is no rule for fin ding. Ther efor e, we must tr avel the r everse p ath and after finding some notable substitution, lo ok for pr oblems to which it c an b e suc c essful ly applie d”. In framework of the Jaco bi metho d we consider H i (2.4) as co nstants of motion for an auxiliary bi-Hamiltonian sys tem on the regular bi-Hamiltonian manifold M a nd treat functionally indep endent eigenv alues u j of N B ( λ ) =  det( N − λ I)  1 / 2 = ( λ − u 1 )( λ − u 2 ) · · · ( λ − u n ) , (2.5) as ”conv enient v ariables ” for an ifinite family of sep ar able bi-integrable s ystems asso ciated with v ar ious separated rela tions Φ i ( u i , p u i , H 1 , . . . , H n ) = 0 , i = 1 , . . . , n , with det  ∂ Φ i ∂ H j  6 = 0 . (2.6) Here u = ( u 1 , . . . , u n ) and p u = ( p u 1 , . . . , p u n ) ar e canonica l v ariables of separa tion { u i , p u j } = δ ij and { u i , p u j } ′ = δ ij u i . (2.7) The Poisson brack ets (2.7) en tail that solutions H 1 , . . . , H n of the sepa rated relatio ns (2.6) ar e func- tionally independent integrals of motion in the bi-involution (2.3), see [26]. Of course, this construction will b e justified only if we are ca pable to obtain separable Hamilton functions H = H i , whic h ha ve natural for m in initial ( p, q ) v aria ble s (2 .1 0). The third constructio n o f integrals of motion in bi-inv olution on ir r egular bi-Hamiltonian manifolds is discuss ed in [18, 31]. In this case po lynomial integrals of motion H 2 , . . . , H n are solutio ns of the following equations for the given Hamiltonian H 1 P ′ dH 1 = κ k P d ln H k , k > 1 , κ k ∈ R , which replace the usual Lenard rela tions (2.4). If this equa tio ns hav e many differ e nt functionally independent solutions la b e led by different κ k , then we obtain so - called sup er integrable s y stems [18, 31]. 2.2 Bi-Hamiltonian structures on cot angen t bundles According to [32] a torsionles s (1,1) tensor field L on a smo oth manifold Q g ives r ise to a (second) Poisson structure on the cotange nt space M = T ∗ Q , compatible with the canonical one. Let θ b e the Liouville 1 -form on T ∗ Q and ω = dθ the standard symplectic 2-for m on T ∗ Q , whose asso cia ted Poisson bivector will b e denoted with P . If w e cho ose some local coordina tes q = ( q 1 , . . . , q n ) on Q and the corres p o nding symplectic co ordina tes ( q , p ) = ( q 1 , . . . , q n , p 1 , . . . , p n ) on T ∗ Q then we get the following loca l express ions θ = p 1 dq 1 + . . . p n dq n , and P =  0 I − I 0  . (2.8) Using a torsionles s tens or field L one can defo r m θ to a 1 -form θ ′ and P to bivector P ′ : θ ′ = n X i,j =1 L ij p i dq j , and P ′ =      0 L ij − L ij n X k =1  ∂ L ki ∂ q j − ∂ L kj ∂ q i  p k      . (2.9) The v a nishing of L tors ion entails that P ′ (2.9) is a Poisson biv ector co mpatible with P . Let us consider natur al integrable by Liouville system on Q . 3 Definition 1 The natur al Hamilton function H 1 = T + V = n X i,j =1 g ij p i p j + V ( q 1 , . . . , q n ) (2.10) is the sum of the ge o desic Hamiltonian T define d by metric tensor g( q 1 , . . . , q n ) and p otential ener gy V ( q 1 , . . . , q n ) on Q . If the corresp onding Hamilton-Jacobi equation is separa ble in or thogonal co or dinate sy stem ( u, p u ) on configura tio nal spa ce Q , then in fra mework of the Eis enhart-Benenti theory the s econd Poisson bivector P ′ (2.9) is defined by a conformal Killing tensor L of gradient type on Q with p oint wise simple eig env alues ass o ciated with the metric g( q 1 , . . . , q n ), see [1, 2, 3, 8, 1 5]. According to Kow alevs ki [17] and Chaplygin [7], separation of v ar iables for integrable systems with higher order integrals of mo tion inv olves generic canonical transfor mation of the whole phase space. Definition (2.10) o f the na tural Hamiltonian and metr ic tenso r g( q 1 , . . . , q n ) is non-inv ar ia nt with res p e c t to arbitrary canonica l transformatio ns of co or dina tes on T ∗ Q q i → q ′ i = f i ( p, q ) , p i → p ′ i = g i ( p, q ) . In the situatio n, when habitual o b jects (geo desic, metric, p otential) los e their geo metric se ns e and remaining inv ar iant equation (2.1) has aprio rity infinite many s olutions, notion of the natura l Poisson bivectors o n T ∗ Q be c ame de-facto very useful practical too l for the ca lculation of v a riables of separation [14, 18, 29, 30, 31, 34]. Definition 2 The natur al Poisson bive ctor P ′ on T ∗ Q is a sum of the ge o desic Poisson bive ct or P ′ T c omp atible with P [ P, P ′ T ] = [ P ′ T , P ′ T ] = 0 , (2.11) and the p otential p art define d by a torsionless (1,1) t ensor field Λ( q 1 , . . . , q n ) on Q P ′ = P ′ T +      0 Λ ij − Λ j i n X k =1  ∂ Λ ki ∂ q j − ∂ Λ kj ∂ q i  p k      . (2.12) In fact, here w e simple as s ume that bi-in tegrability of the geo desic motion is a necess ary condition for bi-integrabilit y in gener ic case at V 6 = 0. Throughout this pap er geo desic bivector P ′ T is de fined by n × n matrix Π( q 1 , . . . , q n , p 1 , . . . , p n ) and functions x , y and z on T ∗ Q P ′ T =         n X k =1  x j k ( q ) ∂ Π j k ∂ p i − y ik ( q ) ∂ Π ik ∂ p j  Π ij − Π j i n X k =1  ∂ Π ki ∂ q j − ∂ Π kj ∂ q i  z k ( p )         (2.13) up to the p oint transfor ma tions. In this case the cor r esp onding Poisson brack et { ., . } ′ lo oks like { q i , p j } ′ = Π ij + Λ ij , { q i , q j } ′ = n X k =1  x j k ( q ) ∂ Π j k ∂ p i − y ik ( q ) ∂ Π ik ∂ p j  , { p i , p j } ′ = n X k =1  ∂ Λ ki ∂ q j − ∂ Λ kj ∂ q i  p k + n X k =1  ∂ Π ki ∂ q j − ∂ Π kj ∂ q i  z k ( p ) . In fact, functions x , y and z are completely deter mined by the matrix Π via compatibility conditions (2.11). 4 W e ca n add v ar ious integrable p otentials V to the given geo desic Hamiltonian T in order to g e t int egrable natur a l Hamiltonians (2.10). In similar manner we can add different co mpatible p otential matrices Λ to the g iven g eo desic matrix Π in order to g et natural Poisson bivectors P ′ (2.12) compatible with the canonical bivector P . Remark 2 W e hav e to underline that this definition of natural Poisson biv ectors is the us eful anzats rather than rigor o us mathematical definition. It is a n obvious sequence o f non-inv a riant definition of the natural Hamilto nia n with respec t to transfo rmations of the whole phase space. W e hope that further inquiry of geometric relations betw een n × n metric matrix g, p otential matr ix Λ and g eo desic matrix Π on T ∗ Q allows us to get more in v ariant and rigor ous definition of these ob jects. Remark 3 In term of v ar iables o f separation Π = 0 a nd Λ = diag( u 1 , . . . , u n ), so w e have usua l inv ariant construction of T uriel [3 2]. The main problem is how to r e w r ite this in v ariant theory in term of initial ph ysical v ar iables. Remark 4 W e suppose tha t (2.13) is a sp ecial fo rm of P ′ T . Other form of P ′ T on t he g eneric s ymplectic leav es of e ∗ (3) for the Steklov-Ly a punov sy stem a t C 2 6 = 0 will b e presented in the forthcoming publication. 3 Sp ecial natural P oisson biv ectors on the sphere The s ta ndard Laplace metho d for the dir ect s earch of integrable s ystems ma y b e applied to the sear ch of the natural bivectors P ′ to o. Firstly we stin t ourselves b y a family of natural Poisson biv ec tors (2.12) with geo desic part (2.13). Then, it is easy to see that the geo desic Hamiltonian T = n X i,j =1 g ij ( q ) p i p j on the co ta ngent bundle T ∗ Q is the second order homogeneous p oly nomial in momenta, so we as sume that entries of Π are the similar homogeneous po lynomials Π ij = n X k,m =1 c km ij ( q ) p k p m (3.1) up to canonical transforma tio ns p k → p k + f k ( q k ). On t wo-dimensional unit sphere Q = S 2 we use spherical c o ordinates (1.3) such that q = ( q 1 , q 2 ) = ( φ, θ ) and p = ( p 1 , p 2 ) = ( p φ , p θ ) . (3.2) A t the third step we in tro duce a family of partial so lutions for w hich all the en tr ies of P ′ T (2.13) are independent on v ariable φ , i.e. at c km ij ( q ) = c km ij ( θ ) , x j k ( φ, θ ) = x j k ( θ ) , y ik ( φ, θ ) = y ik ( θ ) . (3.3) It lo o ks like reaso na ble a ssumption b ecaus e the geo desic Hamiltonia n T T = a 1 J 2 1 + a 2 J 2 2 + a 3 J 2 3 =  a 3 − co t 2 θ ( a 1 sin 2 φ + a 2 cos 2 φ )  p 2 φ , (3.4) − sin 2 φ co t θ ( a 1 − a 2 ) p θ p φ + ( a 1 cos 2 φ + a 2 sin 2 φ ) p 2 θ is indep endent on v aria ble φ at a 1 = a 2 . If a k are co nstants it means that t wo dia gonal elements of inertia tensor of the bo dy a − 1 1 = a − 1 2 are equal to e a ch other and we discuss symmetric rigid bo dy [4]. Due to the s p ec ia l form of P ′ T (2.13) and additional assumptions (3.1-3 .3), equations (2.11) de- comp ose on the subsystem of equatio ns for c km ij ( q ), subsystem of e q uations for z k ( p ) , c km ij ( q ) and third subsystem of equations for x k ( q ) , y k ( q ) , c km ij ( q ), which c a n be pa r tially solved indep endently to each other. 5 Prop ositi on 1 If assumptions (3.1-3.3) hold, then su bsystem of e quations for the fun ctions z k ( p ) c oming in (2.11) has thr e e families of solutions Case 1 . Π ij = 0; Case 2 . z 1 = 0 , z 2 = 0 ; Case 3 . z 1 = p φ 3 , z 2 = p θ 3 . (3.5) This pr op osition gives only the necessary co nditions. O f co urse, there remain co mplement ary e q uations on the other functions c km ij ( θ ), x j k ( θ ) and y j k ( θ ) which hav e to b e s olved in the sequel. A t the first case P ′ T = 0 and w e ca n immediately lo ok for compatible potential part Λ( φ, θ ) and the v ariables of separa tion u 1 , 2 (2.5), which are related with initial v a riables b y the p oint cano nical transformatio ns u i = f i ( φ, θ ) , p u i = g i ( φ, θ ) p φ + h i ( φ, θ ) p θ . (3.6) As a consequence, the geo desic Hamiltonian is a seco nd order homogeneous polyno mia l in ph ysica l a nd separated momenta and the theor y of pro jectively eq uiv alent metrics in class ic a l differen tia l geometry study es s entially the same ob ject [3]. Prop ositi on 2 In se c ond c ase generic solution of (2.11) is p ar ameterize d by six functions g , h and one p ar ameter γ = 0 , 1 : Π =   γ p 2 φ g 1 ( θ ) p 2 φ + g 2 ( θ ) p φ p θ + g 3 ( θ ) p 2 θ 0 h 1 ( θ ) p 2 φ + h 2 ( θ ) p φ p θ + h 3 ( θ ) p 2 θ   (3.7) up to the p oint tr ansformations p k → α k p 1 + β k p 2 . As a bove it is only necess ary condition and functions g , h from (3.7), together with functions x , y fro m (2.13), ar e solutio ns of the remaining s ix non-linea r differential equations in (2.11). Prop ositi on 3 In thir d c ase generic solution of (2.11) is p ar ameterize d by nine functions f , g , h and one p ar ameter γ = 0 , 1 : Π =    f 1 ( θ ) p 2 φ + f 2 ( θ ) p φ p θ + f 3 ( θ ) p 2 θ g 1 ( θ ) p 2 φ + g 2 ( θ ) p φ p θ + g 3 ( θ ) p 2 θ 1 2 f 2 ( θ ) p 2 φ + 2 f 3 ( θ ) p φ p θ + γ  f 3 ( θ ) + h 3 ( θ )  3 / 2 p 2 θ h 1 ( θ ) p 2 φ + h 2 ( θ ) p φ p θ + h 3 ( θ ) p 2 θ    (3.8) up to the p oint tr ansformations p k → α k p 1 + β k p 2 . F unctions f , g , h from (3.8), to g ether with functions x , y fro m (2 .1 3), are s olutions of the remaining 19 non-linear differential equa tions in (2.1 1). Matrices (3 .7) and (3.8) were obtained as so lutions of the subsystem of a lgebraic a nd linear differ- ent ial equations for c km ij ( θ ), which ha s an unambiguous so lutio n. The remaining functions satisfy to the complementary ov erdeter mined subsystem of nonlinear P DE ’s, which ha ve man y distinct par ticular solutions. In both case s (3.7) and (3.8) we can get a complete clas sification o f these par ticular s olutions a nd of the cor resp onding bi-Hamiltonian s ystems (2.4). Classification of separa ble bi-in tegrable systems demands additiona l assumptions on the form of the separated r elations. 3.1 Case 2 - classification of natural bi-Hamiltonian systems Let us briefly dis cuss a pro cedure o f classificatio n of the natural bi-Hamiltonia n sys tems asso ciated with natura l Poisson biv ector (2.1 2-2.13) defined b y the g eo desic ma trix Π (3.7). 6 If h 2 ( θ ) = 0 in (3.7), then six differe nt ial equations coming in (2.11) have four distinct solutions; among them we pic k out so lution defined b y the fo llowing matrix Π =      γ p 2 φ γ 1 − h ′ 3 ( θ ) F α p h 3 ( θ + F 2 ! p φ p θ 0 γ  1 + F 2  p 2 φ + h 3 ( θ ) p 2 θ      , F = tan α Z dθ p h 3 ( θ ) + β ! If a 1 = a 2 = const , then w e ca n put h 3 ( θ ) = γ = 1 without loss of generality and obtain Π =   p 2 φ (1 + tan 2 αθ ) p φ p θ 0 (1 + tan 2 αθ ) p 2 φ + p 2 θ   , y 12 ( θ ) = 2 α x 22 ( θ ) − cos αθ sin αθ α . (3.9) The co rresp onding geo des ic Hamiltonian (2 .4) is equal to T = 1 2 tr N = tr Π = (2 + ta n 2 αθ ) p 2 φ + p 2 θ . A t α = 1 matrix Π (3.9) is consistent o nly with the following p otential matrix Λ =     f ( φ ) g ( φ, θ ) f ′ ( φ ) sin θ 2 cos θ − g ( φ, θ ) 2 cos 2 φ (2 cos 2 θ + 1) g ( φ, θ ) sin 2 φ sin 2 θ + f ( φ ) cos 2 θ + a ta n 2 θ     , (3.10) where f ( φ ) = a cot 2 φ + b sin 2 φ + c sin 2 φ cos 2 φ + 2 d cos 2 φ (2 cos 2 φ − 3) sin 2 φ , g ( φ, θ ) = 2 d sin 3 θ s in 2 φ cos θ . So, bi-Hamiltonia n system ass o ciated with Π (3.9 ) and Λ (3.10) has the following Hamilton function (2.4) H 1 = T + a  ( x 2 1 + x 2 2 ) − x 2 3 ( x 2 1 − x 2 2 )  x 2 1 x 2 3 + (1 + x 2 3 )( x 2 1 + x 2 2 )  bx 2 2 + c ( x 2 1 + x 2 2 )  x 2 1 x 2 2 x 2 3 − 2 d ( x 2 1 + x 2 2 + 2 x 2 1 x 2 3 )  ( x 2 1 + x 2 2 ) − x 2 3 ( x 2 1 − x 2 2 )  ( x 2 1 + x 2 2 ) x 2 1 x 2 3 . Second integral of motion H 2 (2.4) is a fourth or der p olynomial in mo ment a. This integrable system, to the best of o ur knowledge, has not b een considered in liter ature yet. In similar manner we can get a complete class ification of na tural bi-Hamiltonian systems asso ciated with matric e s (3.7) and (3.8). 3.2 Case 3 - one p ossible generalization Non-inv aria nt assumptions (3.1,3.3) dep end on a choice of coo rdinate system a nd we miss a lot of another s o lutions of (2.1), which may be in teresting in applications. One of the pos sible generaliz a tions co nsists in the application of m ultiplica tive separ able functions in (3.1) c km ij ( φ, θ ) = a km ij ( φ ) b km ij ( θ ) , and similar for x , y . F or instance, geo desic matrix Π = e 2i φ    (sin θ p θ + i co s θ p φ ) 2 α 2 p φ (sin θ p θ + i co s θ p φ ) sin 3 θ 0 0    , i = √ − 1 , α ∈ C , (3.11) 7 gives r ise to the na tur al Poisson bivector P ′ at y 11 = − i 2 , z 1 = p φ 3 , z 2 = p θ 3 . It is easy to prove that integrals o f motio n for the Lag range top (4.2) are in in volution with resp ect to the cor resp onding Poisson brack et { ., . } ′ . Remark 5 Bivector P ′ T (2.13) asso ciated with Π (3.11) has a natural counterpart on the generic symplectic le av es o f the Lie algebra e ∗ (3) a t ( x, J ) 6 = 0. 3.3 Case 3 - three-dimensional sphere On the three and four dimensional spheres endow ed with the s tandard spherical co ordinates there are the same three families of solutio ns (3.5). It means that factor 1 / 3 in (3.5) is indep endent o n dimens io n of the sphere. F or instance, if q = ( φ, ψ , , θ ) and p = ( p φ , p ψ , p θ ) ar e the sta ndard spher ical co ordinates on T ∗ S 3 , then at z k = p k 3 matrices Π 1 =         p 2 φ 2 p φ p ψ  4 − 2 f f ′′ f ′ 2  p φ p θ 0 p 2 φ + f p 2 ψ + αf 3 f ′ 2 p 2 θ  2 − 4 f f ′′ 3 f ′ 2  f p ψ p θ 0 2 αf 3 f ′ 2 p ψ p θ p 2 φ − f 3 p 2 ψ + αf 3 f ′ 2 p 2 θ         , f = f ( θ ) , (3.12) Π 2 =   p 2 φ 2 p φ p ψ 2( e αψ + 1 ) p φ p θ 0 F 2 β e − αψ ( e αψ + 1 ) 2 p ψ p θ 0 − 2 γ e − αψ p ψ p θ F − 4 γ ( e − αψ + 1 ) p 2 θ   , where F = ( e αψ + 1) p 2 φ + β e − αψ ( e αψ + 1) 2 p 2 ψ + γ e − αψ p 2 θ , determine geo desic Poisson biv ectors (2.13) and geo des ic Hamiltonians (2.4) T 1 = 3 p 2 φ + 2 f 3 p 2 ψ + 2 αf 3 f ′ 2 p 2 θ , T 2 = (2 e αψ + 3 ) p 2 φ + 2 β ( e αψ + 1 ) 2 e αψ p 2 ψ − 2 γ  2 + 1 e αψ  p 2 θ . Then w e can calculate compatible p otential matr ices Λ 1 , 2 depe nding o n co ordinates ( φ, ψ , θ ) a nd the corres p o nding integrable p o tentials V 1 , 2 . The cor resp onding integrals of motion H 2 , 3 (2.4) are the fourth a nd sixth order p olynomia ls in momenta, resp ectively . So, using notio n of the natura l Poisson bivectors we can pro duce a lot of abstract mathema tica l examples of bi-Hamiltonian system on the spher e. The main problems are how to select physically int eresting bi-Hamiltonia n sy stems and how to construct significant s eparable systems from the non- ph ysical a uxiliary bi-Hamiltonia n systems. 4 Separable bi-in tegrable systems In this Section we presen t matrices Π and Λ fo r the following well-kno w n separ able systems on the sphere • Case 1 - Lag range top, Neuma nn system a nd systems s eparable in the elliptic co o rdinates; • Case 2 - Gor yachev system, Matveev-Dullin system, Kow alevsky top, C ha plygin sys tem; • Case 3 - Goryachev-Chaplygin top, Sok olov system, Kow alevs k y-Goryachev-Chaplygin gyrosta t; 8 which ma y b e na tively e mbedded into the prop o sed s cheme as separable bi- int egrable systems. Some new mathematical genera liz ations of these systems and new separatio n of kno w n sy stems are collateral results for this ac tiv ity . In fra mework of the Jacobi metho ds one gets integrals of motion H 1 , . . . , H n as solutions of the separated r elations (2.6). Of co urse, v a riables o f separ ation and separ ated rela tio ns could hav e the singular points. So, the standa rd pro ble m is the rigor ous determination of domain where v a riables of separatio n and integrals of motion ar e well defined, see the Jacobi definition of the elliptic co ordinates . Our main purpo se is to disc uss natural Poisson bivectors a nd, therefore, we do not comment this hu ge and complica ted pa rt of the work here, see for example [4, 7, 9 , 10, 17, 3 5] and refer ences within. 4.1 Case 1 - Lagrange top If the spherica l co or dinates φ, θ (1.3 ) a re v a riables of separ ation, one gets the simplest natural P oisson bivector P ′ (2.13) at Π = 0 , and Λ =  φ 0 0 θ  . (4.1) The a ux iliary bi-Hamiltonian system is trivial H 1 = φ + θ , H 2 = 1 2 ( φ 2 + θ 2 ) . On the other ha nd, s ubstituting v a riables of separation u 1 = φ and u 2 = θ into the separ ated relations Φ 1 =  a + cos 2 θ sin 2 θ  H 2 − H 1 + p 2 θ + b c os θ = 0 , Φ 2 = p 2 φ − H 2 = 0 , one gets in tegrals of motion for the Lagr ange top in rotating fra me H 1 = J 2 1 + J 2 2 + aJ 2 3 + bx 3 , H 2 = J 2 3 , a, b ∈ R , (4.2) More complicated natural bi-vector P ′ obtained fro m matrix (3.11) gives rise to another v ariables of separatio n for this sys tem. Remark 6 According to [27 ] bivector P ′ (2.12) asso ciated with Λ (4.1 ) admits extensio n from cotan- gent bundle T ∗ S 2 to the symplectic leaves of the Lie algebra e ∗ (3) a t ( x, J ) 6 = 0. 4.2 Case 1 - Neumann system Let us put P ′ T = 0 in (2.12) and consider some particular solution P ′ of the equations (2 .1) defined by the following non-sy mmetric matrix Λ =      a 1 cos 2 φ + a 2 sin 2 φ ( a 1 − a 2 ) sin 2 φ 2 cos θ sin θ ( a 1 − a 2 ) sin 2 φ 2 cos θ s in θ a 3 sin 2 θ + ( a 1 sin 2 φ + a 2 cos 2 φ ) co s 2 θ      (4.3) with three arbitrar y parameters a k ∈ R . As ab ove, the auxilia r y bi-Hamiltonian system has trivia l int egrals of motion H k (2.4), which are functions only on the c onfiguratio na l s pace S 2 . On the other ha nd, co ordinates of separation u j (2.5) are the standa rd elliptic co ordinates on the sphere x 2 1 λ − a 1 + x 2 2 λ − a 2 + x 2 3 λ − a 3 = ( λ − u 1 )( λ − u 2 ) ( λ − a 1 )( λ − a 2 )( λ − a 3 ) . (4.4) By substituting these v a riables in the separ ated rela tions u i H 1 − H 2 − 4( a 1 − u i )( a 2 − u i )( a 3 − u i ) p 2 u i + U i ( u i ) = 0 , i = 1 , 2 , 9 one gets bi-in tegrable systems with q uadratic in moment a integrals of mo tion H 1 = J 2 1 + J 2 2 + J 2 3 + V ( x ) , H 2 = a 1 J 2 1 + a 2 J 2 2 + a 3 J 2 3 + W ( x ) , which ar e in the bi-inv olution (2.3) with r esp ect to both P oisson brack ets. Here V ( x ) a nd W ( x ) are easy ca lculated from the p otentials U 1 , 2 . F o r instance, if U ( u ) = u ( u − a 1 − a 2 − a 3 ) , then one gets the Neumann system with the following integrals of motion H 1 = J 2 1 + J 2 2 + J 2 3 + a 1 x 1 + a 2 x 2 + a 3 x 3 , H 2 = a 1 J 2 1 + a 2 J 2 2 + a 3 J 2 3 − a 2 a 3 x 1 − a 1 a 3 x 2 − a 1 a 2 x 3 . (4.5) Remark 7 Bivector P ′ (2.12) as so ciated with Λ (4.3) also sa tisfies equatio ns (2.1) at ( x, J ) 6 = 0, but in this case w e lo se bi-inv olutivity (2.3) o f integrals of motion H 1 , 2 (4.5) for the C le bs ch system on the whole phase space e ∗ (3). Of course, the corresp o nding elliptic co ordinates on e ∗ (3) r emain v ar iables of se pa ration, but w e can not get interesting natura l Hamiltonians using these v ariables [25]. 4.3 Case 2 - systems with cubic in tegral of motion A t γ = 0 in (3 .7 ) we hav e pa r ticular s o lution of the equations (2.11) defined b y geo des ic matrix Π =     0 − i 2  ∂ ∂ θ + 2 h ( θ ) g ( θ )  F 0 F     , F =  g ( θ ) p θ − i h ( θ ) p φ  2 , i = √ − 1 , (4.6) depe nding on a rbitrar y functions g ( θ ) a nd h ( θ ) and by functions x 22 = − g ( θ ) 2 h ( θ ) , y 12 = 0 , z k = 0 . This ma tr ix Π is consistent with the diagonal p otential matrix Λ = α ex p  i φ − Z h ( θ ) g ( θ ) dθ    1 0 0 1   . (4.7) The corr esp onding bi-Hamiltonian s ystems (2.4) are non-physical T = F and, therefore, we immedi- ately procee d to consideration of the co or dinates of separ a tion v 1 , 2 = √ u 1 , 2 following to [34]. If we int ro duce p oly nomial B ( λ ) = ( λ − v 1 )( λ − v 2 ) = λ 2 − i √ F λ + Λ 1 , 1 . instead o f characteristic po lynomial B ( λ ) = ( λ − u 1 )( λ − u 2 ) = ( λ − v 2 1 )( λ − v 2 2 ) (2 .5) of recursio n op erator N , then it is eas y to prove that {B ( λ ) , A ( µ ) } = λ µ − λ  B ( λ ) λ − B ( µ ) µ  , {A ( λ ) , A ( µ ) } = 0 , where A ( λ ) = Z i dθ g ( θ ) − i p φ λ . It entails that p v j = A ( λ = v j ) , j = 1 , 2 , 10 are ca nonically conjuga ted to v j momenta and that the cor resp onding Poisson brack ets r ead as { v i , p v j } = δ ij , { v i , p v j } ′ = δ ij v 2 i . Now we hav e to subs titute this family of v a riables of separation into the s eparated relations and try to get natural Hamiltonia ns. F or instance, let us ta ke g ( θ ) = sin θ f ( θ ) , h ( θ ) = cos θ f ( θ ) , (4.8) substitute λ = v j µ = 2i 3 v j p j , j = 1 , 2 , int o the equation Φ( λ, µ ) = µH 1 + H 2 − µ 3 − λ 3 − bλ + α 2 λ = 0 , (4.9) and solve a pair of the resulting eq uations with res pe ct to H 1 , 2 . If a 1 = a 2 in the geo desic Hamiltonian (3.4), then in this solution we hav e to put f ( θ ) = cos 1 / 3 θ sin 2 θ , and we obtain integrals of motion for the Gorychev system on the spher e [12] H 1 = J 2 1 + J 2 2 + 4 3 J 2 3 + 2i αx 1 x 2 / 3 3 − b x 2 / 3 3 , (4.10) H 2 = 2 J 3 3 J 2 1 + J 2 2 + 8 9 J 2 3 − b x 2 / 3 3 ! − 2 i αx 1 / 3 3 J 1 + 4i α 3 x 2 / 3 3 x 1 J 3 . F or other separable na tur al bi-in tegrable s ystems from [23, 34] w e pr esent Hamiltonians and functions g and h only . So, for the Gor yac he v -Chaplygin top [11, 7] we ha ve H 1 = J 2 1 + J 2 2 + 4 J 2 3 + ax 1 + b x 2 3 , g ( θ ) = 1 cos θ s in θ , h ( θ ) = 3 cos 2 θ − 2 cos 2 θ s in 2 θ . F or the Dullin-Matveev system [9] with Hamiltonia n H 1 = J 2 1 + J 2 2 +  1 + x 3 x 3 + c − x 2 3 − | x | 2 4( x 3 + c ) 2  J 2 3 + ax 1 ( x 3 + c ) 1 / 2 + b x 3 + c geo desic ma trix Π (4.6) and p o tent ial ma trix Λ (4.7) a r e defined by functions g ( θ ) = 1 sin θ , h ( θ ) = − 1 − 2 c co s θ − 3 co s 2 θ 2 sin 2 θ (co s θ + c ) . F or the system with the Hamiltonia n H 1 = J 2 1 + J 2 2 +  7 12 + x 3 2( x 3 + | x | )  J 2 3 + 2i αx 1 ( x 3 + | x | ) 5 / 6 − b ( x 3 + | x | ) 1 / 3 , bi-Hamiltonian str uc tur e is defined by functions g ( θ ) = (cos θ + 1) 2 / 3 sin θ , h ( θ ) = − (cos θ + 1) 2 / 3 2(cos θ ) − 1) . F or the last sys tem fro m [23] we hav e H = J 2 1 + J 2 2 +  13 16 + 3 x 3 8( x 3 + | x | )  J 2 3 + ax 1 ( x 3 + | x | ) 3 / 4 + b ( x 3 + | x | ) 1 / 2 11 and g ( θ ) = (cos θ + 1) 1 / 2 sin θ , h ( θ ) = (3 cos θ + 1)(cos θ + 1) 1 / 2 4 sin 2 θ . If e P ′ is the linear in momenta Poisson bivector fro m [34], then o ur natural P oisson bivector is e qual to P ′ = e P ′ P − 1 e P ′ . Remark 8 According to [33] , the Cor yachev-Chaplygin, Chaplygin and Dullin-Matveev systems can be embedded into a family of integrable sy s tems with cubic in tegral of motion. W e suppo se that bi- Hamiltonian structures for the V alent sys tems may be des c rib ed by a suitable choice of the functions g ( θ ) and h ( θ ) in (4.6) and (4.7). Remark 9 Another p oss ible gener alization c o nsists of multiplication of matrix Π (4 .6) on the functions depe nding on φ similar to (3.11). 4.4 Case 2 - Kow alevski top and Chaplygin system Let us consider a geo desic bivector P ′ T (2.13) deter mined by the matrix Π Π = 1 sin α θ cos 2 θ    0 2 p φ p θ α 0 cos 2 θ p 2 φ + sin 2 θ p 2 θ    , α ∈ R , (4.11) and by functions y 12 ( θ ) = cos θ  sin θ + α x 22 ( θ ) co s θ  , z 1 , 2 = 0 . There is only one po tent ial matrix consistent with Π (4.11) Λ =     a cos αφ − b sin αφ  a sin αφ − b cos αφ  cot θ  a sin αφ − b co s αφ  tan θ − a cos αφ + b sin αφ     , a, b ∈ R . (4.12) The co rresp onding co or dinates of s eparatio n u 1 , 2 (2.5) are the ro ots of the p olynomial B ( λ ) = λ 2 − p 2 θ sin 2 θ + p 2 φ cos 2 θ sin α θ c os 2 θ λ − ( a cos αφ − b sin αφ )( p 2 θ sin 2 θ + p 2 φ cos 2 θ ) sin α θ c os 2 θ (4.13) − 2 sin θ ( a sin αφ + b cos αφ ) p φ p θ sin α θ c os 2 θ − a 2 − b 2 . F ollowing to [29, 30] we can introduce auxiliary p olyno mia l A ( λ ) = sin θ p θ α cos θ λ + a sin αφ + b cos αφ α p φ − sin θ ( a cos αφ − b sin αφ ) α cos θ p θ , such a s { B ( λ ) , A ( µ ) } = 1 λ − µ  ( µ 2 − a 2 − b 2 ) B ( λ ) − ( λ 2 − a 2 − b 2 ) B ( µ )  , { A ( λ ) , A ( µ ) } = 0 . It entails that p u j = − 1 u 2 j − a 2 − b 2 A ( λ = u j ) , j = 1 , 2 , are the ca nonically c onjugated moment a satisfying to the Poisson brackets (2.7). A t α = 2 these v ariables have b een consider ed b y Cha plygin [6]. 12 By s ubstituting these v ariables o f separa tion int o a pair o f the s eparated relations Φ 1 = ( u 2 1 − a 2 − b 2 ) p 2 u 1 + H 1 − H 2 = 0 , Φ 2 = ( u 2 2 − a 2 − b 2 ) p 2 u 2 + H 1 + H 2 = 0 , one gets separable bi-integrable sys tem with the Hamilton function 2 α 2 H 1 = p 2 φ − tan 2 θ p 2 θ + 2 ( a cos αφ + b cos αφ ) c o s α θ , α ∈ R . (4.14) According to [29, 30], at α = 1 us ing separ ated relations Φ( u, p u ) =  ( u 2 − a 2 − b 2 ) p 2 u + H 1 − H 2  ( u 2 − a 2 − b 2 ) p u + H 1 + H 2  + cu 2 + du = 0 (4.15) one gets Hamilton function of the generaliz e d Kow alev ski top [17] H kow = 2 H 1 =  1 − c + 1 x 2 3  ( J 2 1 + J 2 2 ) + 2 J 2 3 + 2 ax 2 + 2 bx 1 − d p x 2 1 + x 2 2 . (4.16) A t α = 2 w e can use another separated rela tions Φ( u, p u ) =  ( u 2 − a 2 − b 2 ) p 2 u + cu + H 1 − H 2  ( u 2 − a 2 − b 2 ) p 2 u + cu + H 1 + H 2 ) + du = 0 (4.17) in or der to get Hamiltonian o f the generalized Chaplyg in system [6 , 1 3] H ch = 8 H 1 =  1 − 4 c + 1 x 2 3  ( J 2 1 + J 2 2 ) + 2 J 2 3 − 2 a ( x 2 1 − x 2 2 ) − 2 bx 1 x 2 − 2 d 1 + 4 c − x 2 3 . (4.18) A t c = − α − 2 we hav e geo des ic Hamiltonian T = J 2 1 + J 2 2 + 2 J 2 3 with the constant inertia tensor. Remark 10 By substituting these v a riables o f sepa ration into a nother separation relations we can obtain v ar ious mathematica l genera lizations of bi-integrable Hamiltonians (4.14,4.16,4.18). 4.5 Case 2 - spherical top and Chaplygin system A t γ = 0 in (3 .7 ) we hav e a particular solution of the equations (2.11) defined b y matrix Π =      p 2 φ α − sin 2 θ cos 2 θ s in 2 θ p φ p θ 0 α sin 2 θ p 2 φ + α − sin 2 θ cos 2 θ p 2 θ      , α ∈ R , (4.19) and functions y 12 = sin θ cos θ + 2 α cos 2 θ sin 2 θ − α x 22 , z k = 0 . In this case co or dinates of separation u 1 , 2 (2.5) are equal to u 1 = p 2 φ , u 2 = αp 2 φ sin 2 θ − (sin 2 θ − α ) p 2 θ cos 2 θ , so that conjugated momenta read as p u 1 = arctan  p θ tan θ p φ  − φ 2 p φ , p u 2 = sin θ cos θ ar c tan   sin 2 θ p θ q α cos 2 θp 2 φ − sin 2 θ (s in 2 θ − α ) p 2 θ   2 q α cos 2 θp 2 φ − sin 2 θ (s in 2 θ − α ) p 2 θ . 13 By substituting these v a riables of separatio n into the sepa rated r elations Φ 1 = √ u 1 − H 2 = 0 , Φ 2 = αH 1 − u 2  1 − ( α − 1 ) tan 2 (2 p u 2 √ u 2 )  + αf ( θ ) = 0 , where θ = a rccos s u 2 − αH 2 2 u 2 + α ( H 2 2 − u 2 )(1 − cos 4 p u 2 √ u 2 ) 2 u 2 one gets generalized L a grang e top with integrals of mo tion H 1 = J 2 1 + J 2 2 + J 2 3 + f ( x 3 ) , H 2 = J 3 . (4.20) Other sepa rated relations Φ 1 ( u 1 , p u 1 ) = 2 √ u 1 sin  4 p u 1 √ u 1  H 2 − H 1 + u 1 = 0 , (4.21) Φ 2 ( u 1 , p u 1 ) = αH 1 − u 2  1 − ( α − 1) tan 2 (2 p u 2 √ u 2 )  = 0 . give rise to integrals of motion for the spherica l top H 1 = T = J 2 1 + J 2 2 + J 2 3 , H 2 = J 1 J 2 J 3 . (4.22) There ar e only t wo p otential matrices co mpatible with Π (4.19) Λ (1) =     f ( φ ) 0 f ′ ( φ )(sin 2 θ − α ) 2 sin θ co s θ αf ( φ ) sin 2 θ     and Λ (2) =      a sin 2 φ + b cos 2 φ − cos θ α sin θ ( α − sin 2 θ )( a co s 2 φ − b sin 2 φ ) − sin θ α cos θ ( α − sin 2 θ )( a co s 2 φ − b sin 2 φ ) − ( α − 2 sin 2 θ )( a s in 2 φ + b cos 2 φ ) α      . In the first cas e the auxilia ry bi-Hamiltonian system with the Hamilton function (2 .4) H (1) 1 =  1 + α − 1 x 2 3   J 2 1 + J 2 2  + 2 J 2 3 + f  x 1 x 2   1 + α x 2 1 + x 2 2  , (4.23) is a deformation of the geo desic Hamiltonian fo r the Kow alevski to p a t α = 1 and f = 0. By substituting the corre s p o nding co ordina tes o f sepa ration (2.5) ˆ u 1 = u 1 + f ( φ ) , ˆ p u 1 = p u 1 − 1 2 Z φ dx p 2 φ + f ( φ ) − f ( x ) , ˆ u 2 = u 2 + αf ( φ ) sin 2 θ , ˆ p u 2 = p u 2 , int o Φ 1 = ˆ u 1 − b H 2 = 0 and the sec ond separated relation Φ 2 in (4.21), one gets a ge neralizatio n of the spherical to p defined b y the following integrals of mo tion b H 1 = J 2 1 + J 2 2 + J 2 3 + f  x 1 x 2  x 2 1 + x 2 2 , b H 2 = J 2 3 + f  x 1 x 2  x 2 1 + x 2 2 + x 2 3 . In the second ca se matrices Π (4.1 9) and Λ (2) give rise to the auxiliary bi-Ha miltonian system with the Hamilton function H (2) 1 =  1 + α − 1 x 2 3   J 2 1 + J 2 2  + 2 J 2 3 + 4 α − 1 ax 1 x 2 − 2 α − 1 b ( x 2 1 − x 2 2 ) . (4.24) It is a new deformation of the well-known Chaplyg in system [6 ]. 14 Remark 11 According to [21], there is a no n-canonica l map, which relates int egrals of motion (4.2 2) with integrals of motion for the Gaffet s y stem [10] H 1 = J 2 1 + J 2 2 + J 2 3 − 1 ( x 1 x 2 x 3 ) 2 / 3 , H 2 = J 1 J 2 J 3 + x 2 x 3 J 1 + x 1 x 3 J 2 + x 1 x 2 J 3 ( x 1 x 2 x 3 ) 2 / 3 . In order to descr ibe the bi-Hamiltonian structure for the Gaffet s ystem w e hav e to use additional non-p oint trans formation o f the standard spher ical co ordinates, which changes the form o f P ′ T (2.13) in initial v ariables . This bi- Ha miltonian structure will b e discussed in the for thcoming publicatio n. 4.6 Case 3 - Goryac hev-Chaplygin top and Sok o lov system A t γ = 0 in (3.8) equations (2 .1 1) hav e a par ticular solution P ′ T (2.13) defined by the following symmetric matrix Π =   p 2 θ + p 2 φ (4 + 3 cot 2 αθ ) 2 p φ p θ 2 p φ p θ p 2 θ − p 2 φ cot 2 αθ   , α ∈ R , (4.25) and by the functions x 22 = y 12 = − cos αθ sin αθ α , z k = p k 3 . There is only one po tent ial matrix compatible with Π (4 .25) Λ = a cos 2 αθ  1 0 0 1  . The co rresp onding auxiliar y bi-Hamiltonian system is defined by the Hamilton function (2.4) 1 2 H 1 = (2 + cot 2 αθ ) p 2 φ + p 2 θ + a cos 2 αθ . If α = 1, w e have a defo r mation of the g eo desic Hamiltonian for the Kow a levski top [17] 1 2 H 1 = J 2 1 + J 2 2 + 2 J 2 3 + a x 2 3 . (4.26) This a uxiliary bi-Hamiltonia n system g ives rise to the v ariables o f sepa ration u 1 , 2 (2.5) u 1 , 2 =   p φ ± s p 2 φ sin 2 θ + p 2 θ + a cos 2 θ   2 =  J 3 ± r J 2 1 + J 2 2 + J 2 3 + a x 2 3  2 , α = 1 . A t a = 0 these co ordina tes were found in [7]. By substituting the generaliz ed Chaplygin v ar iables v 1 , 2 = √ u 1 , 2 , p v 1 , 2 = − 1 2i ln  v 1 , 2 (i x 1 − x 2 ) − (i J 1 − J 2 ) x 3  + ln( v 2 1 , 2 − a ) 4i , (4.27) int o the separated r elations Φ 1 , 2 ( v , p v ) = H 1 v + H 2 + b p v 2 − a s in 2 p v − v 3 − cv 2 = 0 , v = v 1 , 2 , p v = p v 1 , 2 , one gets in tegrals of motion for the gener a lized Gor yachev-Chaplygin gyr ostat [11, 7] H 1 = J 2 1 + J 2 2 + 4 J 2 3 + 2 cJ 3 + bx 1 + a x 2 3 H 2 = (2 J 3 + c )  J 2 1 + J 2 2 + a x 2 3  − bx 3 J 1 . 15 By substituting the same v ariables (4.27) into the following separated relations Φ 1 , 2 ( v 1 , 2 , p v 1 , 2 ) = b H 1 ± b H 2 + b q v 2 1 , 2 − a sin 2 p v 1 , 2 − v 2 1 , 2 − cv 1 , 2 = 0 , (4.28) we o btain the genera lized Sokolo v s y stem [20] defined by integrals of motion b H 1 = J 2 1 + J 2 2 + 2 J 2 3 + cJ 3 + b ( J 3 x 1 − x 3 J 1 ) + a x 2 3 , b H 2 =  2 J 3 + c + bx 1  r J 2 1 + J 2 2 + J 2 3 + a x 2 3 , up to the canonica l transfor mation discussed in [16]. 4.7 Case 3 - Kow alevski-Gory ach ev-Chaplygin gyrostat The g e o desic matrix Π (4.25) fo r the Goryac hev-Chaplygin top may be deformed b Π = Π + β    0 cos αθ sin 3 αθ p 2 φ 0 0    , (4.29) if y 11 ( θ ) = x 21 ( θ ) − β 2 α , y 12 ( θ ) = − cos 2 αθ sin 2 αθ x 22 ( θ ) − cos αθ α sin αθ , z k = p k 3 . Remark 12 In the r -matrix formalism transition from ma trix (4.2 5) to the matrix (4.29) generates transition fro m the quadratic Sklyanin bracket to the so-called re fle c tion equation algebra [22, 28]. In ge ne r ic case matr ix b Π (4.29) is compatible with the p otential matrix b Λ (1) = a e − 4 αφ β sin 2 αθ   cos 2 αθ − 4 − β cos αθ sin αθ 4 cos αθ sin αθ β cos 2 αθ   + b sin 2 αθ cos 2 αθ   1 − β sin αθ cos αθ 0 1   (4.30) The co rresp onding auxiliar y bi-Hamiltonian system is defined by the Hamiltonia n 1 2 H 1 = (2 + cot 2 αθ ) p 2 φ + p 2 θ + a (cos 2 αθ − 2)e − 4 αφ β sin 2 αθ + b sin 2 αθ cos 2 αθ . So, at α = 1 we hav e a nother deformation of the geo desic Hamiltonian for the Kow a levski top [17] 1 2 H 1 = J 2 1 + J 2 2 + 2 J 2 3 − a ( x 2 1 + x 2 2 + 1 ) x 2 1 + x 2 2 e − 4 arctan( x 1 /x 2 ) β + b ( x 2 1 + x 2 2 ) x 2 3 . In this case des c ription of the v a r iables of separation and the co rresp onding bi-int egrable system is an op en pr oblem. A t β = ± 2 i there is one more particula r p o tential matrix co mpa tible with b Π (4.29) b Λ (2) = γ e ± i αφ   ± i sin αθ cos αθ 0 0   . (4.31) In this par ticular case we can substitute the co ordinates of sepa r ation u 1 , 2 (2.5) and the co rresp o nding momenta p u 1 , 2 int o the separated relatio ns defined b y Φ( u, p u ) = u 6 + H 1 u 4 + H 2 u 2 + a + p b ( u ) sin 2 p u = 0 , (4.32) 16 and obtain int egrals of motio n for the Kow a levski-Gor yac hev -Chaplygin gyrostat with the following Hamilton function H 1 = J 2 1 + J 2 2 + 2 J 2 3 + 2 c 1 J 3 + c 2 x 1 + c 3 ( x 2 1 − x 2 2 ) + c 4 x 2 3 , (4.33) see [6, 13, 17, 35]. Here b ( u ) (4.32) is a specia l p oly nomial of eigh t order in u with co efficients dep ending on a and c k , see details in [22]. Remark 13 In this ca se in order to get the conjuga ted momenta p u 1 , 2 and the separated relation we used the Lax matrices a nd the reflection equation algebra , that dra s tically simplified all the c alculations. Remark 14 F or the systems with quartic integral of motion from [24] the natura l P o isson bivector may be obtained using deformation of the matr ix (4.29) similar to (3.11). 4.8 Case 2 - deformations of the Kow alevski top and Chaplygin systems Let us consider triv ia l canonica l transformatio n p θ → p θ + f ( θ ) , (4.34) which pr eserved canonical Poisson bivector P (2.8). This mapping shifts the natural Poisson bivector P ′ (2.12) as so ciated with matrices Π (4.11) and Λ (4.12) by the r ule b P ′ = P ′ + g ( θ )                0 0 0  α cos 2 θ − 1 α sin 2 θ + cot θ α ln ′ g  p φ ∗ 0 0 p θ + cos 2 θ sin α − 2 θ 4 g ( θ ) ∗ ∗ 0 sin α − 2 θ ( a s in αφ + b cos αφ ) 2  sin θ cos θ ln ′ g − 1  ∗ ∗ ∗ 0                , where g ( θ ) = − 2 f ( θ ) sin 2 − α θ cos 2 θ and ln ′ g = 1 g ( θ ) dg ( θ ) dθ . The Poisson bivector b P ′ gives r ise to the ”shifted” v ariables o f separation ˆ u = u | p θ → p θ + f ( θ ) , ˆ p u = p u | p θ → p θ + f ( θ ) . (4.35) If we substitute these v aria bles of separ ation in to the o ld sepa rated rela tio ns (4.15) and (4.17) o ne gets non-natural Hamiltonians, which a re related to the old Ha miltonians (4.16) and (4.18) by canonical transformatio n (4.34). In order to g et new natural Hamiltonians w e hav e to appropriately modify the separated relations. F or insta nce, let us take f ( θ ) = √ β tan α − 1 θ cos α θ . A t α = 1 b y substituting v ariables o f sepa ration (4.35) into the new separa ted relatio ns b Φ = Φ − β H 1 + β 2 + p β ( ˆ u 2 − a 2 − b 2 ) ˆ p u , ˆ u = ˆ u 1 , 2 , ˆ p u = ˆ p u 1 , 2 , (4.36) where Φ is given by (4.15), one g ets g eneraliza tio n o f the Hamilton function (4.16) b H kow =  1 − c + 1 x 2 3  ( J 2 1 + J 2 2 ) + 2 J 2 3 + 2 ax 2 + 2 bx 1 − d p x 2 1 + x 2 2 − β x 3 , 17 A t α = 2 the ”shifted” separa ted relations b Φ = Φ + p β ( ˆ u 2 − a 2 − b 2 ) ˆ p u , ˆ u = ˆ u 1 , 2 , ˆ p u = ˆ p u 1 , 2 , (4.37) where Φ is given by (4.17), yield similar gener alization of the Hamiltonian (4 .18) b H ch =  1 − 4 c + 1 x 2 3  ( J 2 1 + J 2 2 ) + 2 J 2 3 − 2 a ( x 2 1 − x 2 2 ) − 2 bx 1 x 2 − 2 d 1 + 4 c − x 2 3 + β  1 x 4 3 − 1 x 6 3  . These Hamiltonia ns at c = − α − 2 and ano ther Hamiltonians asso c ia ted with v ar ious functions f ( θ ) may be found in [35]. The se pa rability o f these systems, to the best of our knowledge, has not b een considered in liter- ature yet. In tb oth cas e s equations of motion ar e linearized on the t w o copies of the non- hyperelliptic curves of g enus three defined by (4.3 6) and (4.37). W e do not know how to solv e the corres po nding Abel- J acobi e q uations a s yet. Remark 15 Other natural Poisson bivectors studied in the previous Sectio ns may b e s hifted on the similar linear in momenta terms. As ab ov e, it allows us to get v ario us generaliza tions of the considered bi-integrable systems. 5 Conclusion W e prov ed that a lmost a ll known in tegrable sy s tems on the tw o -dimensional unit sphere S may b e studied in the framework of a single theory of natur al Poisson biv ec tors. It is an ex p er imental fact suppo rted b y a ll the know co nstructions o f the v ar iables of separa tion on the sphere. 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