A new approach to separation of variables for the Clebsch integrable system. Part I: Reduction to quadratures
aa r X i v : . [ n li n . S I] F e b A new approach to separation of variables for the Clebschintegrable system. Part I: Reduction to quadratures
Y. Fedorov , F. Magri , T. Skrypnyk , Polytechnic University of Catalonia, Barcelona, Spain Dipartimento di Matematica e Applicazioni- Universit´a di Milano Bicocca, Milano, Italia Universit´a degli Studi di Torino, via Carlo Alberto 10, 10123, Torino, Italia Bogolyubov Institute for Theoretical Physics, Metrolohichna str.14-b, 03115, Kiev, Ukraine [email protected], [email protected], [email protected]
Abstract
The paper describes a new concept of separation of variables with a concrete application tothe Clebsch integrable case of the Kirchhoff equations. There are two principal novelties: Thefirst is that the separating coordinates are constructed (not guessed) by solving the Kowalewskiseparability conditions. The second is that the solutions of the equations of motion are written interms of theta-functions by means of a generalization of the standard Jacobi inversion problem ofalgebraic geometry. These two novelties are dealt with in two separate parts of the paper. Part Iexplains the Kowalewski separability conditions and their implementation to the Clebsch case. Itis shown that the new separating coordinates lead to quadratures involving Abelian differentialson two different non-hyperelliptic curves (of genus higher than the dimension of the invariant tori).In Part II these quadratures are interpreted as a new generalization of the standard Abel–Jacobimap, and a procedure of its inversion in terms of theta-functions is worked out. The theta-functionsolution is different from that found long time ago by F. K¨otter, since the theta-functions used inthis paper have different period matrix.
Keywords: algebraic integrable systems, separation of variables, Abelian varieties, Jacobi inversion prob-lem.
The famous Clebsch integrable case of the Kirchhoff equations [1] belongs to an ample class ofclassical and modern algebraic integrable systems, for which finding a separation of variables is anespecially difficult problem. The class also includes, for instance, the Euler-Frahm top on the algebra so (4), integrated by Schottky [2], the Kowalewski top, the Henon-H´eiles system, and the geodesic flowon SO (4) [3, 4]. For the general case of motion, the Clebsch case has been first solved in theta-functions1y F. K¨otter [5], who devised a reduction to quadratures involving a pair of points on a genus 2hyperelliptic curve (the so called K¨otter’s curve). The coordinates of the points give the separatingcoordinates, and the quadratures themselves have the form of a standard Abel–Jacobi map. Later on,the theta-function solutions for the Clebsch case and the Euler–Frahm top have been reconstructed ina series of publications, including [6, 7], where the method of Lax representation and of the relatedBaker–Akhiezer functions [8] was used instead of separation of variables. Furthermore, the algebraicgeometric structure of the complex invariant tori, and the linearization of the flow from the standpointof the theory of Abelian varieties, have been described in detail in [9, 10].The present paper aims to revise the approach based on the method of separation of variables. Ourapproach is unconventional both for the technique used to find the separating coordinates, and for thetechnique used to solve the equations of motion in terms of theta-functions. The first difference is thatto find the separating coordinates we utilize the technique of the Kowalewski separability conditions[11, 12, 13]. This means that the separating coordinates are systematically obtained (not guessed) bysolving the above conditions in each specific example. The second difference is that our quadraturesinvolve sums of Abelian integrals on two different non-hyperelliptic curves C and K of genus 3 (while thequadratures of K¨otter involve a single hyperelliptic curve of genus 2). They lead to a new generalizationof the Abel-Jacobi map. This unexpected feature does not forbid, however, to invert the map and towrite the solutions of the equations of motion in terms of theta-functions. These novelties are terselydiscussed below.The Kirchhoff equations describing the motion of a rigid body in an ideal fluid can be written inthe form ˙ ~S = ~S × ∂H∂ ~S + ~T × ∂H∂ ~T , ˙ ~T = ~T × ∂H∂ ~S , where ~S and ~T are vectors in the three-dimensional Euclidean space E , and H is a scalar function.These equations can be interpreted as a Hamiltonian vector field X H on the dual to the Lie algebra ofthe group of motions of E . The Clebsch integrable case corresponds to the choice H = X α =1 m α S α + X α =1 n α T α , where the coefficients satisfy the constraint n − n m + n − n m + n − n m = 0 . It is convenient to represent these coefficients in the form m α = σ + τ j α n α = σ ( j β + j γ ) + τ j β j γ , σ, τ ∈ R , ( α, β, γ ) = (1 , , ,j , j , j being arbitrary parameters; and to regard H as the linear pencil H = σH p + τ H s generated bythe Hamiltonians H p = X α =1 S α + ( j β + j γ ) T α , H s = X α =1 j α S α + j β j γ T α . (1)2n the sequel we assume that j , j , j are distinct.By our definition, the Clebsch system is the pair of Hamiltonian vector fields X p and X s associatedwith the above Hamiltonians and given by X p ( S α ) = ( j γ − j β ) T β T γ ,X p ( T α ) = S β T γ − S γ T β (2)and, respectively, X s ( S α ) = j α ( j γ − j β ) T β T γ + ( j β − j γ ) S β S γ ,X s ( T α ) = j β S β T γ − j γ S γ T β . (3)It is known that the vector fields X p and X s commute, and that they possess four integrals of motion,namely the above Hamiltonians H p and H s , as well as the functions C = X α =1 T α S α , K = X α =1 T α . (4)The procedure for explicitly solving these differential equations consists of two stages. First, weintroduce a suitable set of separating coordinates. Namely, in the complex projective space P withhomogeneous coordinates ( A α , B α ), α = 1 , ,
3, consider the intersection of four quadrics given byequations X α =1 ( j β + j γ ) B α + A α = 0 , X α =1 j β j γ B α + j α A α = 0 , X α =1 A α B α = 0 , X α =1 B α = 0 , ( α, β, γ ) = (1 , , . (5)Since the equations are independent, the intersection is a curve E ⊂ P . Its intersection with thehyperplane X α =1 A α T α + B α S α = 0 (6)gives eight points, which move along E while the coordinates S α , T α vary in time with the fields X p and X s . With each point we associate the complex parameter v = − P α =1 j α A α P α =1 A α . (7)Let v p and v s denote its derivatives along the vector fields X p and X s respectively. Then each intersectionpoint will give us the pair of coordinates x = v , x = − v s v p . (8)3he eight pairs of ( x , x ) constructed in this way have the remarkable property to verify the samesystem of differential equations. Namely, let g ( x ) = K x + H p x + H s , and let dx = x p dt p + x s dt s , dx = x p dt p + x s dt s be the differentials of x , x , where ( x p , x p ) and( x s , x s ) denote their derivatives with respect to the vector fields X p and X s , and t p and t s are the timeparameters along these fields. Then the system reads dx w ( w − g ( x )) + √ dx W ( W − g ( x )) = iC dt s ,x dx w ( w − g ( x )) + √ x dx W ( W − g ( x )) = iC dt p , i = √− . (9)Here the the pairs of the variables ( x , w ) and ( x , W ) satisfy equations of algebraic curves C : w − g ( x ) w + g ( x ) − C ( x + j )( x + j )( x + j ) = 0 ,K : W − g ( x ) W + 4 C ( x + j )( x + j )( x + j ) = 0 . (10)These are the genus 3 non-hyperelliptic curves C and K mentioned above. In the integral form, theequations (9) give the quadratures Z PP (cid:18) ω ω (cid:19) + Z RR (cid:18) ¯ ω ¯ ω (cid:19) = (cid:18) t s t p (cid:19) , P = ( x , w ) ∈ C, R = ( x , W ) ∈ K, (11)where ( ω , ω ) and (¯ ω , ¯ ω ) are certain holomorphic differentials on the curves C and K respectively,and P , R are some base points on them.The second stage of our procedure is the inversion of the quadratures. The facts that the genusof the curve is higher than the dimension of the invariant tori and that the periods of the Abelianintegrals on the two curves are distinct, raise a natural question about the invertibility of (11) in termsof meromorphic functions of the complex times t p , t s . Nevertheless, by using the results of [9, 10, 14],we show that the quadratures describe a well-defined map P from C × K to the two-dimensional Prymsubvariety of the Jacobian of K . We call it the Abel–Prym map. To our best knowledge, such kind ofmaps did not appear before, neither in the classical nor in the modern literature. Moreover, in contrastto the standard Abel–Jacobi map, which is one-to-one almost everywhere, under the Abel–Prym mapeach point of the Prym variety has 8 preimages ( P , R ) , . . . , ( P , R ) on C × K . They can be foundas zeros of a higher order theta-function Ξ ∗ which is quasiperiodic on the Prym variety, that is, byapplying a generalization of the Riemann vanishing theorem, which we formulate and prove.As a result, any symmetric functions of the coordinates of ( R , . . . , R ) (or of ( P , . . . , P )) can bewritten in terms of Ξ ∗ and its derivatives. On the other hand, the coordinates of these preimages coincidewith the eight sets of separating variables found initially from the point ( S α , T α ) in the phase space.The variables ( S α , T α ) can be written in terms of symmetric functions of the eight pairs of separatingcoordinates. Combining these results enables one to obtain theta-function solutions for the Clebschintegrable case. 4art I of the paper deals with the first stage: construction of separation variables and reduction toquadratures. Section 2 introduces the Kowalewski separability conditions in the context of the theoryof Poisson pencils and bi-Hamiltonian geometry. In Section 3 these conditions are used to compute theseparating coordinates for the Clebsch system. These coordinates are studied in Section 4, where thequadrature formula shown before is explicitly worked out. Finally, Section 5 contains short commentson potential extensions of the technique developed in the paper. The appendix contains the proof ofthe Main Proposition stated in Section 3.Part II deals with the second stage: analysis of the main properties of the Abel-Prym map generatedby the quadratures (11) and the algorithm of its inversion in terms of theta-functions of higher order,which lead to a new theta-function solution of the Clebsch system. This part consists of three sections.The two parts are kept separate because each of them has its own intrinsic interest, independentof the example of the Clebsch system. They correspond to different but complementary ways of ap-proaching the study of separable systems: the differential-geometric viewpoint of the theory of Poissonmanifolds, and the algebraic-geometric viewpoint of the theory of Abelian varieties. Also the languagesused in the two parts are quite different. These peculiarities suggest to deal with them in separate form.Notwithstanding, the two parts must be considered together, because both of them are necessary toprovide a full picture of the behavior of the solutions of the Clebsch equations of motion. In this section we introduce a particular class of integrable Hamiltonian vector fields and we in-vestigate their separability properties. The class contains the Clebsch system. The initial point of theinvestigation is the concept of Poisson pencil. The final point is a particular form of the Kowalewskiseparability conditions. They allow to recover, quite rapidly, the particular solution of the Clebsch sys-tem obtained by Weber in 1878 [15]. The analysis of this elementary example will serve as guideline forthe general study of the Clebsch system carried out in the next section.Let M be a smooth manifold of dimension m , and P : T ∗ M → T M , Q : T ∗ M → T M be a pair ofbivectors defined on M (that is, second-order skew-symmetric contravariant tensor fields). Assume that P and Q verify the Schouten conditions[ P, P ] Sch = 0 , [ P, Q ] Sch = 0 , [ Q, Q ] Sch = 0 , where [ · , · ] Sch denotes the Schouten bracket on the exterior algebra of multivectors. Then M is calleda bihamiltonian manifold, and the pencil of bivectors Q − λP , λ being a real or complex parameter,is called a Poisson pencil. The interest for these pencils is due to their ability of generating integrableHamiltonian systems through their polynomial Casimir functions [16]. Let g ( λ ) = K λ q + K λ q − + · · · + K q +1 be a polynomial with coefficients in the ring of differentiable functions on M , and let dg ( λ ) = dK λ q + dK λ q − + · · · + dK q +1
5e its differential. The function g ( λ ) is said to be a polynomial Casimir function of the pencil Q − λP if( Q − λP ) dg ( λ ) = 0for any value of λ . In this case, the first coefficient K is a Casimir function of P , the last coefficient K q +1 is a Casimir function of Q , and the intermediate coefficients satisfy the recursive relations QdK a = P dK a +1 . There are q of such relations. Each of them defines a vector field X a = QdK a = P dK a +1 , for a =1 , . . . , q . Thus a polynomial Casimir function of degree q endows M with q + 1 distinguished functions( K , K , . . . , K q +1 ) and with q distinguished vector fields ( X , X , . . . , X q ). Assume that Q − λP has n polynomial Casimir functions of degrees ( q , q , . . . , q n ), and suppose that the dimension m of themanifold, the number n of polynomial Casimir functions, and the sum p = q + · · · + q n of the degreesof the polynomial Casimir functions fulfill the relation m = n + 2 p . Then, it is a well known result ofthe theory of Poisson pencils that each vector field defined by the pencil is an integrable Hamiltoniansystem in the sense of Liouville. Let us see how this works for the Clebsch system. Example 1.
On two copies of the Euclidean space E , equipped with coordinates ( S α , T α ), considerfour bivectors: P = X α S α ∂∂T β ∧ ∂∂T γ ,P α = S α ∂∂S β ∧ ∂∂S γ + ( T β ∂∂S γ − T γ ∂∂S β ) ∧ ∂∂T α , ( α, β, γ ) = (1 , , . Any pair of them verifies the Schouten conditions defining a Poisson pencil, and, therefore, any linearcombination of P , P , P , P with constant coefficients is again a Poisson bivector. Consider the bivectors Q = P + j P + j P + j P , P = P + P + P . They endow M with the structure of a bihamiltonian manifold. The corresponding Poisson pencil Q − λP has two polynomial Casimir functions. The first has degree zero, g ( λ ) = C . The second has degree two: g ( λ ) = K λ + K λ + K . Thus the Poisson pencil defines two vector fields X and X . Since the relation m = n + 2 p is verified,they form an integrable bihamiltonian hierarchy. Working with the coordinates ( S α , T α ) one may checkeasily that the functions ( C , K , K , K ) and the vector fields ( X , X ) coincide with the functions( C , K , H p , H s ) and the vector fields ( X p , X s ) of the previous section (possibly up to an irrelevant com-mon constant multiplicative factor). Hence one may claim that the Clebsch system is the bihamiltonianhierarchy defined by the Poisson pencil discussed in this example. (cid:3) provided by the pencil itself . To discuss thisproblem we have recourse to the theory of Kowalewski separability conditions, from which we extractthe following particular result. Proposition 1
Let X p and X s be a pair of commuting vector fields on a manifold M of dimension m ,possessing ( m − independent integrals of motion ( H , . . . , H m − ) . Let furthermore h ( x ) = Ex + F x + G be a quadratic polynomial whose coefficients satisfy the Kowalewski separability conditions EF s − F E s = EG p − GE p , EG s − GE s = F G p − GF p , (12) where the symbols E p , F p , G p , E s , F s , G s denote the derivatives of the coefficients of h ( x ) along the vectorfields X p and X s respectively. Then the roots x and x of the quadratic equation Ex + F x + G = 0 (13) are separating coordinates for X p and X s .Proof. Let F be the 2-dimensional foliation spanned by the vector fields X p and X s . The functions H , . . . , H m − are constant on the leaves of the foliation. Let us add to these functions the roots x and x of the polynomial Ex + F x + G . Together they form a coordinate system on M adapted tothe foliation. In view of the Kowalewski conditions and the V i ` ete formulas x + x = − F/E and x x = G/E , one easily finds that the roots x and x satisfy the differential equations x s + x x p = 0 , x s + x x p = 0 . (14)Consider then the vector fields X s + x X p and X s + x X p . The previous relations entail that these vectorfields commute, and that X s + x X p is a multiple of ∂/∂x , while X s + x X p is a multiple of ∂/∂x .Hence, there exist two functions ψ and ψ of the coordinates ( x , x , H , . . . , H m − ) such that ψ ∂∂x = X s + x X p ψ ∂∂x = X s + x X p . (15)By the commutativity of X s + x X p and X s + x X p , the function ψ does not depend on x , whereas ψ does not depend on x . This is a key property in the process of separation of variables.To proceed further, assume that the multipliers ψ and ψ are algebraic functions of x and x respectively, that is they are rational on certain algebraic curves C ⊂ C ( x , w ) and K ⊂ C ( x , W ).Thus one may set ψ = R ( x , w ) and ψ = R ( x , W ). In terms of differential forms, the vectorequations (15) then read dx R ( x , w ) + dx R ( x , W ) = dt s x dx R ( x , w ) + x dx R ( x , W ) = dt p , (16)7here, as above, t p , t s are the time parameters along X p , X s respectively.If the functions R , R coincide, the above quadratures represent a kind of Abel–Jacobi map (stan-dard or generalized), which can be inverted in terms of the Riemann theta-functions or their gener-alization. However, if R , R are different and the corresponding curves C and K are not birationallyequivalent, the problem of inversion of (16) may be more delicate or even not solvable: for the case ofthe Clebsch system it will be addressed in the second part of the paper. (cid:3) To relate the theory of Poisson pencil with the Kowalewski separability conditions we recall thatany Poisson bivector P defines a derivation d P on the exterior algebra of multivectors. It is defined by d P = [ P, · ] Sch , has degree 1, and verifies the cohomological condition d P = 0. It allows to define thespecial class of vector fields Z for which Z ∧ d P Z = 0 on the whole manifold M or, at least, on a levelsurface of the Casimir functions of P . Here is an example of such a vector field. Example 2.
Consider again the Poisson pencil Q − λP of the previous example, and the vector field Z = P α T α ∂∂T α . The Lie derivative of P along Z vanishes, while the Lie derivative of Q is Lie Z Q = − X α =1 S α ∂∂T β ∧ ∂∂T γ . Since d P ( Z ) = Lie Z P , one finds that Z ∧ d P Z = 0 Z ∧ d Q Z = − C ∂∂T ∧ ∂∂T ∧ ∂∂T . Therefore the 3-vectors Z ∧ d P Z and Z ∧ d Q Z vanish on the submanifold C = 0. To interpret thisresult properly, notice that Z ( C ) = 2 C . Therefore Z ( C ) is again a Casimir function, and one maysay that the 3-vectors Z ∧ d P Z and Z ∧ d Q Z vanish on the submanifold Z ( C ) = 0. (cid:3) Let us expand the above example, by introducing the following class of vector fields.
Definition 2
Let M be a bihamiltonian manifold of dimension m = 2 q + r + 1 endowed with a Poissonpencil Q − λP with r common Casimir functions ( C , C , . . . , C r ) , and with a single polynomial Casimirfunction g ( λ ) of degree q . A vector field Z such that: • The derivatives Z ( C a ) of the common Casimir functions are still common Casimir functions • The 3-vectors Z ∧ d P Z and Z ∧ d Q Z vanish on the submanifold defined by the equations Z ( C a ) = 0 will be referred to as a field verifying the Frobenius conditions with respect to the Poisson pencil Q − λP (this terminology is borrowed from the theory of differential forms, where the 1-form α is said to verifythe Frobenius condition if α ∧ dα = 0 ). For our purposes it is sufficient to deal with the case q = 2. The following Proposition shows that thevector fields that satisfy the Frobenius conditions are relevant to the theory of separation of variables.8 roposition 3 Let M be a bihamiltonian manifold of dimension m = 5 + r , endowed with a Poissonpencil Q − λP having r common Casimir functions C a and a polynomial Casimir function g ( x ) ofdegree . Let X p and X s be the bihamiltonian hierarchy defined by the Poisson pencil (possibly up to anirrelevant common constant multiplicative factor). Assume that there exists a vector field Z satisfyingthe Frobenius conditions. Then the roots of the derivative of g ( x ) along Z , h ( x ) = Z ( g ( x )) , (17) are separating coordinates for X p and X s on the invariant submanifold S , defined by the equations Z ( C a ) = 0 .Proof. The value of the 3-vector Z ∧ d P Z on the differentials of the coefficients K , K , K of g ( x ) isthe scalar function X cyclic Z ( K )( Z { K , K } P − { Z ( K ) , K } P − { K , Z ( K ) } P ) , where the sum is over the cyclic permutations of (1 , , K is a Casimir function of P and K , K commute with respect to P , many terms in the above expression vanish, and one obtains( Z ∧ d P Z )( dK , dK , dK ) = E ( − F s + G p ) + F E s − GE p . Let us restrict this equation to the invariant submanifold S , and notice that the restriction of thederivatives of E, F, G coincide with the derivatives of the restriction of the functions, because the vectorfields X p and X s are tangent to S . As a result, the functions E, F, G satisfy the first Kowalewskiseparability condition on S provided that Z verifies the Frobenius condition on S . Under replacementof P by Q , the same argument shows that E, F, G satisfy the second Kowalewski separability conditionas well. (cid:3)
It follows that the Poisson pencils which are endowed with a vector field verifying the Frobeniusconditions may be qualified as separable pencils , because they provide the separating coordinates fortheir bihamiltonian hierarchies. The following example, which is closely related to the Clebsch system,is quite instructive.
Example 3.
The vector field Z of Example 2 verifies the Frobenius conditions with respect to thePoisson pencil Q − λP of Example 1 on the invariant submanifold { Z ( C ) = 2 C = 0 } . In this case thepolynomial Casimir function is g ( x ) = X α =1 ( x + j β )( x + j γ ) T α + ( x + j α ) S α . According to Proposition 3, the coefficients of the quadratic polynomial h ( x ) = 2 X α =1 ( x + j β )( x + j γ ) T α C = 0. Thus the roots of the algebraicequation X α =1 T α x + j α = 0 . are separating coordinates for the Clebsch system on { C = 0 } . They were first used by H. Weber ([15])in 1878. The coordinates of Weber are, therefore, a simple outcome of the Kowalewski separabilityconditions. (cid:3) Before proceeding to our construction of separating variables for the Clebsch system, we make a lastdigression to the theory of Poisson pencils, namely to a process which may be called ”lift of a Poissonpencil”. It will be presented here in a form adapted to our example.Let M be the phase space of the Clebsch system, endowed with the Poisson pencil Q − λP , and let E be an elliptic curve given in parametric form. The coordinates of points of E are elliptic functionsof a parameter ζ varying on a torus C /L , L being a period lattice. We do not specify initially neitherthese functions nor the lattice. The reason is that the choice of the curve E is irrelevant to the processof lifting, while it will become crucial later in the search of the separating coordinates. At that point,it will be suggested by the separability conditions.To lift the Poisson pencil of the Clebsch system (Example 1) to the extended phase space ˆ M = M ×E ,let us choose a scalar function ˆ C on ˆ M , and let π : ˆ M → M be the canonical projection. Note thatthere exists a unique pair of bivectors ˆ P and ˆ Q on ˆ M having the following properties: • They can be projected from ˆ M onto M along π . • Their projections are the bivectors P and Q on M . • The function ˆ C is a common Casimir function of ˆ P and ˆ Q .The simplest way to see these properties is to equip the extended phase space with the fibered co-ordinates ( S α , T α , ζ ) adapted to the projection, and to define the bivectors ˆ P and ˆ Q by giving thefundamental Poisson brackets of the coordinate functions on ˆ M . The Poisson brackets of the coordi-nates ( S α , T α ) are the same as on M : { S α , S β } ˆ P = { S α , S β } P { S α , T β } ˆ P = { S α , T β } P { T α , T β } ˆ P = { T α , T β } P . The Poisson brackets of ( S α , T α ) with the new coordinate ζ are { S α , ζ } ˆ P = − X β =1 ∂ ˆ C ∂S β { S α , S β } P + ∂ ˆ C ∂T β { S α , T β } P ! ∂ ˆ C ∂ζ ! − { T α , ζ } ˆ P = − X β =1 ∂ ˆ C ∂S β { T α , S β } P + ∂ ˆ C ∂T β { T α , T β } P ! ∂ ˆ C ∂ζ ! − . C to be a Casimir function. The Poisson brackets of ˆ Q aredefined similarly. This explicit representation allows to check the above claims, and to recognize thatthe new pencil ˆ Q − λ ˆ P inherits all the properties of the Poisson pencil on M . In particular, • The pencil ˆ Q − λ ˆ P is a new Poisson pencil, called the lift of Q − λP from M to ˆ M . • It has two common Casimir functions ˆ C and ˆ C , and a quadratic polynomial Casimir functionˆ g ( λ ) = ˆ K λ + ˆ K λ + ˆ K . • The coefficients of the polynomial Casimir function ˆ g are π -related to the coefficients of thepolynomial Casimir function g of Clebsch. Thus the two sets of functions coincide in the fiberedcoordinates ( S α , T α , ζ ). The same claim is true for the Casimir function ˆ C : it coincides with thefunction C in such coordinates.The Casimir function ˆ C is the function used to implement theprocess of lifting. • The new Poisson pencil defines two vector fields ˆ X p and ˆ X s on ˆ M , which are integrable sinceˆ m = ˆ n + 2ˆ p . • The vector fields X p and X s of the Clebsch system are the π -projections of the fields ˆ X p and ˆ X s .As a result, by lifting the Poisson pencil Q − λP from M to ˆ M one obtains an integrable extensionof the Clebsch system for any choice of the curve E and for any choice of the Casimir function ˆ C .This class of lifts, however, is too ample for our purposes. We are interested in a lifted Poisson pencilwhich possesses a vector field ˆ Z satisfying the Frobenius conditions on the submanifold ˆ S defined bythe equations ˆ Z ( ˆ C ) = 0 and ˆ Z ( ˆ C ) = 0. This will enable one to implement the Kowalewski separabilityconditions on ˆ S . Furthermore, we want ˆ S to be a six-dimensional covering space of the base manifold M . Only in this case one can hope to use the separating coordinates provided by the Kowalewskiseparability conditions on ˆ S as separating coordinates on the entire phase space M . Of course, in thisprocess one must take care of the multi-valuedness of the canonical projection π restricted to ˆ S . Thisproblem will be discussed later on.A realization of the above program requires appropriate choices of the function ˆ C and of the vectorfields ˆ Z . To choose ˆ Z we consider a class of fields depending uniquely on the parameter ζ of E . Thisallows us to reduce the complicated systems of PDE’s, coming from the Frobenius conditions, to a moretractable system of ODE’s. Hence we setˆ Z = X α =1 A α ( ζ ) ∂∂S α + B α ( ζ ) ∂∂T α , (18)so that ˆ Z does not act on the additional coordinate ζ . To choose ˆ C , we recall that the Frobeniusconditions require that Z ( ˆ C ) and Z ( ˆ C ) be again Casimir functions. Our choice is to identify ˆ C with Z ( ˆ C ) and to set Z ( ˆ C ) = 0. Accordingly, we setˆ C = X α =1 B α ( ζ ) S α + A α ( ζ ) T α , X α =1 A α B α = 0 . (19)11n this way the invariant submanifold ˆ S is a six-dimensional covering space of the base manifold M ,defined by the single equation ˆ C = 0.As a result, the pencil ˆ Q − λ ˆ P , the vector field ˆ Z , and the invariant submanifold ˆ S are uniquelydefined by six functions A α ( ζ ) and B α ( ζ ). Our aim is to choose these functions in such a way thatˆ Z satisfies the remaining Frobenius conditions on ˆ S . The existence of these functions is stated in thefollowing proposition proved in Appendix 1. Proposition 4
Assume that A α ( ζ ) and B α ( ζ ) verify the four quadratic constraints (5) listed in theIntroduction. Then the vector field ˆ Z , defined by (18) , satisfies the Frobenius conditions with respect tothe Poisson pencil ˆ Q − λ ˆ P on the invariant submanifold ˆ S . This result is the keystone of our construction of separating coordinates for the Clebsch system. Itwill be fully exploited in the next section. Before, we use Proposition 4 to specify the elliptic curve E ,the invariant submanifold ˆ S , and the canonical projection π : ˆ S → M .First, note that the functions A α ( ζ ) , B α ( ζ ), verifying the constraints (5), are rational on a spatialcurve in C ( v , v , v ) given by the equations v − v = j − j , v − v = j − j . (20)Since j , j , j are distinct, the above curve is elliptic. Then one can write A α = c α v β v γ , B a = c α v α , (21)assuming that the coefficients c α satisfy the constraints P α =1 c α = 0 , P α =1 j α c α = 0. So c α = ρ ( j α − j β )( j α − j γ ) , where ρ is an arbitrary nonzero factor. Since all the equations of the present theory are gauge-invariantwith respect to the choice of the gauge function ρ , one may assume, without loss of generality, that ρ is a constant. In the sequel we will choose ρ = 1. The conclusion is that one can identify E with theabove elliptic curve, proving the claim that E is fixed by the separability conditions.Next, the submanifold ˆ S ⊂ ˆ M is fully specified as the zero-locus of the Casimir function ˆ C . Byinserting the above parametric representations of A α and B α into the first equation in (19), one obtains X α =1 c α ( v β v γ T α + v α S α ) = 0 . (22)Jointly with (21), (20), this equation defines ˆ S . Finally, the restriction of the projection π onto thesubmanifold ˆ S have the following property, which follows from Proposition 4. Corollary 5
The invariant submanifold ˆ S is a 8-fold covering of the base manifold M . roof. The curve E can be transformed to the canonical Weierstrass form in C ( Z, Y )ˆ E : Y = 4( Z + j )( Z + j )( Z + j ) (23)by means of the birational transformation v α = ( Z + j α )( Z + j β ) − ( Z + j β )( Z + j γ ) + ( Z + j γ )( Z + j α ) Y . (24)Indeed, under (23), substituting (24) into (20) converts the latter into identities.Next, substituting (24) into (22) yields P ( Z ) Y + P ( Z ) = 0 , where P ( Z ) and P ( Z ) are certain polynomials of degree 2 and 4 respectively, with coefficients depend-ing linearly on ( S α , T α ). (These polynomials will be presented in explicit form in the second part of thepaper.) Eliminating from here Y by using (23) gives a polynomial equation of degree eight in Z . Hence,to any given point of M there correspond eight points on ˆ S . (cid:3) From a different perspective, this corollary justifies the claim, made in the Introduction, that thecurve in P defined by the four quadrics (5) is cut by the plane (6) in eight points. Indeed, the previousargument shows that there is a one-to-one correspondence between the intersections in P of the quadrics(5) with the plane (6) and the inverse images on ˆ S of a point in M . From this point of view it is interestingto interpret, within the geometry of the lift of the Poisson pencil, the meaning of the parameter v associated by (7) to any intersection point in P . A simple substitution of the parametric equations of A α and B α into (7) shows that v is the function v = v α − j α , (25)associated to anyone of the eight inverse images of ( S α , T α ) on ˆ S . This remark gives us the chanceto summarize the part of the study concerning the lifting process and the Kowalewski separabilityconditions in ˆ M as follows. Each point ( S α , T α ) of the phase space M of Clebsch define eight pointson the submanifold ˆ S , and therefore eight values of the function v . By the Kowalewski separabilityconditions the function v is one of the separating coordinates of the extensions ˆ X p and ˆ X s of theClebsch system (as we shall see in the next section). Therefore, each solutions of the Clebsch systemon M selects eight solutions of the separation equations for ˆ X p and ˆ X s on ˆ S . The interesting propertyis that this correspondence may be inverted: from each suitable set of eight solutions of the separationequations of ˆ X p and ˆ X s on ˆ S one can reconstruct a solution of the Clebsch equations of motion. Thisdelicate mechanism will be studied in the second part of the paper, where it will be shown that one canexploit such a 8:1 correspondence to write the solutions of the Clebsch equations of motion in termsof theta-functions. Before addressing this reconstruction problem, however, one has still to identify theseparating coordinates and the form of the equations of the vector fields ˆ X p and ˆ X s in these coordinates,at least on ˆ S . This is the task of the last section of this first part.13 Separation of variables in action
As we know, on the extended space ˆ M = M × E the vector field ˆ X p is characterized by two mainproperties: • Its projection onto M is the vector field X p of the Clebsch system. • The function ˆ C is one of its integral of motion.We also know that ˆ X p is an integrable bihamiltonian vector field, and that its restriction to the invariantsubmanifold ˆ S = { ˆ C = 0 } is separable. This means that in the coordinate system formed by thepolynomial Casimir functions and by the roots x , x of the derivative (17) of the polynomial Casimirfunction, the restriction of ˆ X p to ˆ S admits the expansion (coming from Eq. 15)( x − x ) ˆ X p = ψ ∂∂x − ψ ∂∂x , (26)where the component ψ does not depend on x , and ψ does not depend on x . In this section wecompute these components explicitly and, using them, bring the differential equations defined by thefields ˆ X p , ˆ X s to quadratures. A change of coordinates.
To reach this goal it is convenient to use the following redundant coordi-nate system on ˆ M . We will keep using the coordinates v , v , v parameterizing the points of the fiber E and introduce six new coordinates X α = B α S α = c α v α S α , Y α = A α T α = c α v β v γ T α . (27)Then the equation of ˆ S ⊂ ˆ M and of the field ˆ Z take the following simple formsˆ S : X α =1 ( X α + Y α ) = 0 , ˆ Z = X α =1 c α (cid:18) ∂∂X α + ∂∂Y α (cid:19) , (28)whereas the expressions of the polynomial Casimir functions becomeˆ C = 1 v v v X α =1 d α X α Y α , ˆ C = X α =1 ( X α + Y α ) , ˆ g ( x ) = X α =1 d α ( x + j β )( x + j γ )( v + j β )( v + j γ ) Y α + X α =1 d α ( x + j α )( v + j α ) X α , d α := 1 c α . The function v has been defined in (25). As shown before, it coincides with the function v defined byformula (7) in Introduction. 14 he vector field ˆ X p . There are infinitely many vector fields on ˆ M whose projection onto M give X p , they depend on an arbitrary function. The whole class of these fields is specified by the equations˙ v α = 1 v α ξ , ˙ X α = 1 v α X α ξ + iv v v ( j γ − j β ) Y β Y γ , ˙ Y α = v β + v γ v β v γ Y α ξ + iv v v ( j γ − j β )( v γ X β Y γ − v β X γ Y β ) , where ξ is an arbitrary scalar function on ˆ M . The first equation is due to the equations defining thecurve E . The second equation is obtained in three steps. First, one substitutes ˙ S α by the correspondingdifferential equations of the field X p into the identity ˙ X α = ˙ B α S α + B α ˙ S α ; then, one replaces ( S α , T α )by the new coordinates ( X α , Y α ); finally, one exploits the definitions of the functions ( A α , B α ) and ofthe parameters c α given in the previous section. The third equation is obtained in a similar way. Amongthe vector fields of this class, ˆ X p is characterized by the property of leaving the function ˆ C invariant.This condition yields ξ = i v v v U/V , where U = X α =1 ( j β − jγ )( v γ X β Y γ − v β X γ Y β − ( j β − jγ ) Y β Y γ ) V = X α =1 v β v γ X α + v α ( v β + v γ ) Y α . (29)Restrictions of U, V onto the submanifold ˆ S describe the restriction of the field ˆ X p . In this connection,one observes that the restriction of U admits the following factorization U (cid:12)(cid:12) ˆ S = LM, L = X α =1 ( x + j α ) Y α , M = X α =1 j α ( X α + Y α ) ; (30)hence the function ξ simplifies to ξ = i v v v M L/V , and the components of ˆ X p along the curve E become ˆ X p ( v α ) = iv β v γ M LV . (31)These expressions play a key role in the process of separation of variables.
Separating coordinates.
In the new coordinate system ( v α , X α , Y α ), α = 1 , ,
3, the equation (13),whose roots give the separating coordinates x , x according to Proposition 3, takes the following sym-metric form X α =1 ( x + j α )( v + j α ) X α + ( x + j β )( x + j γ )( v + j β )( v + j γ ) Y α = 0 . (32)15ne of its roots, say x , is the function v introduced above, and x is related to x by a Mobiustransformation, namely x = v, x = ax + bcx + d , (33)where a = X α =1 j α X α , b = − X α =1 j β j γ X α + j α ( j β + j γ ) Y α , c = X α =1 Y α , d = X α =1 j α Y α . The first relation in (33) holds because setting x = v reduces the equation (32) to the constraint (28).The second relation follows from the Vi´ete formulas. Note that on ˆ S the determinant of the Mobiustransformation ad − bc = X α =1 ( j α − j β )( j α − j γ ) X α Y α is proportional to the first Casimir function ˆ C . Thus this transformation is regular on ˆ C = 0 outsideof the singular variety ˆ C = 0. Next, the determinant, which is a quadratic form of the coordinates, onˆ S factorizes into the product of two linear factors: ad − bc = LN, N = X α =1 ( x + j α ) X α on ˆ S . (34)Other relevant property is that on ˆ S the difference x − x is the rational function x − x = VL , (35)the linear forms L and V have been defined in (30), (29). These observations are sufficient to reducethe Clebsch system to quadratures. The components ψ and ψ . According to (26), the components ψ , ψ of the field ˆ X p can be writtenas ψ = ( x − x ) x p , ψ = ( x − x ) x p . Thus, to compute them one has to find first x p , x p . The formeris easy to find: x p = v p = 2 v α v αp = 2 i v v v M LR .
Due to (35), the above relation immediately yields ψ = 2 i v v v M .In this way we obtain ψ as a function of the redundant coordinates ( X α , Y α , v α ). To express it interms of the separating coordinates, that is, as a function of the polynomial Casimir functions and of x , we use the following (easily checked) identity M = ˆ g ( x ) + 2 v v v ˆ C and the identity v v v = p ( x + j )( x + j )( x + j ).16etermination of x p is longer and computationally more challenging. To compute it one might derivethe Mobius transformation (33) along ˆ X p , then insert the derivatives of x , X α , Y α into the resultingexpression, and take into account the constraint (28). Next, one needs to replace ( X α , Y α ) by theirexpressions in terms of the separating coordinates. In this form the computation cannot be brought toan end due to overwhelming complexity of the resulting expressions (containing thousands of terms).The way to bypass this difficulty is suggested by the study of the polynomial Casimir functions ofthe Poisson pencil. As we have seen, on ˆ S p Φ( x ) ˆ C = LN, ˆ g ( x ) + 2 p Φ( x ) ˆ C = M , Φ( x ) := ( x + j )( x + j )( x + j ) . These factorization properties suggest introducing a new coordinate system on ˆ S including the separa-tion coordinates x , x and the linear forms L, M, N . The system is completed by choosing, as a sixthcoordinate, X := P ( x + j β )( x + j γ ) X α .The transformation ( X α , Y α ) → ( x , x , L, M, N, X ) is given by the five equations above defining x , L, M, N, X , and by the constraint equation (28). Solving them with respect to X α , Y α , one obtainsthe expressions of the inverse transformation d α X α = ( x + j α )( x + x + j β + j γ ) x − x ( L − M + N ) + ( x + j α ) N + X ,d α Y α = ( x + j β )( x + j γ ) x − x ( L − M + N ) + ( x + j α ) L + X . (36)The use of the new coordinates drastically simplifies computation of x p , which is based on the equation x p = x p − LV p − V L p L = x p + ( x − x ) L p L − V p L .
The derivatives L p , V p of the linear forms L, V along ˆ X p are still rather complicated, but after theirinsertions into the above equation an impressive cancellation of terms takes place, which leads to thefollowing simple relation ( x − x ) x p = i Φ( x ) L p Φ( x ) − p Φ( x ) L N ! . Therefore ψ = i Φ( x ) L p Φ( x ) − p Φ( x ) L N ! . To complete the computation of x p , it remains to express L, N in terms of the polynomial Casimirfunctions. This is performed by using the third noticeable relationˆ g ( x ) = Φ( x )Φ( x ) L + N , which is obtained in the same way as x p was. 17 eduction to quadratures. The above expressions for ψ and ψ may be recast in a more symmetricform by rescaling them by the factor ˆ C . Indeed, a simple algebraic manipulation yieldsˆ C ψ = i M ( M − ˆ g ( x )) , ˆ C ψ = i N (ˆ g ( x ) − N ) . Setting here w = M and W = −√ N , we arrive at the final form:ˆ C ψ = iw ( w − ˆ g ( x )) , ˆ C ψ = i W √ W − ˆ g ( x )) . As a consequence of the above expressions, the new variables w, W are manifestly related to the constantsof motion and the separating coordinates as described in equations (10). The latter define the pair ofalgebraic curves C and K associated with the Clebsch system by the theory of Kowalewski separabilityconditions.Finally, to bring the Clebsch system to quadratures we plug the above representation of ψ and ψ into the Abelian form of the equations of motion (16) provided by the theory of Kowalewski separabilityconditions. The result is given by the equations (9) in Introduction. Their integration leads to thequadrature formula (11). A description of its properties and its inversion will be given in the secondpart of the paper. Reconstruction formulae.
We now express the original variables S α , T α on the phase space M interms of the separating variables ( x , w ) , ( x , W ).Let us recall that the Poisson tensor P , described in Example 1, has two Casimir functions C and K , and that its symplectic leaves are, accordingly, four-dimensional submanifolds of M . Consider thegeneric symplectic leaf { C = e = 0 , K = f } and notice that the inverse image of it with respect tothe projection π : ˆ S → M is the four-dimensional submanifold of ˆ S satisfying the conditions ˆ C = e ,ˆ K = f . In the new coordinates ( x , x , L, M, N, X ) on ˆ S this submanifold is defined by equations e = LN p Φ( x ) , f = L Φ( x ) ( L (2 x + x + j + j + j ) + 2 X ) + ( L − M + N ) ( x − x ) . (37)Solving this system with respect to L, X , one obtains them as functions of x , x , M, N, e, f .On the other hand, recall that the projection π : ˆ S → M is described by equations (36), which, inview of (27), can be written in the form S α = c α (cid:18) √ x + j α ( x + x + j β + j γ ) x − x ( L − M + N ) + p x + j α N + X √ x + j α (cid:19) ,T α = c α p ( x + j β )( x + j γ ) x − x ( L − M + N ) + √ x + j α p ( x + j β )( x + j γ ) L + X p ( x + j β )( x + j γ ) ! . (38)Replacing here L, X by the solutions L ( x , x , M, N, e, f ), X ( x , x , M, N, e, f ) of (37), then setting, asabove, M = w , N = − W/ √
2, we arrive at algebraic expressions for S α , T α in terms of ( x , w ) , ( x , W ),on the symplectic leaf { C = e, K = f } . (Their explicit form is not too long, but we prefer not topresent it here.) Combined with solutions (inversion) of the quadratures (9), these expressions give ussolutions of the Clebsch system. 18 emark. The reconstruction formulae (38) give S α , T α as rational functions of x , p Φ( x ) , w , whichare meromorphic on the curve C in (10), and of x , W , which are meromorphic on K . However, theseformulae involve also the radicals √ x + j α , α = 1 , ,
3, which are not meromorphic on C , but on its4-fold unramified covering D → C , D being a genus 9 curve whose properties are described in detail inPart II of the paper. As a result, for a generic pair of points ( x , w ) ∈ C , ( x , W ) ∈ K , the formulae(38) yield not one but 4 sets ( S α , T α ), which, however, are only different by signs.Indeed, taking squares of the right hand sides of (38), we obtain rational functions of x , p Φ( x ) , w and x , W . Thus, S α , T α are determined by a pair of points ( x , w ) ∈ C , ( x , W ) ∈ K uniquely.Finally, observe that, by their construction, expressions (38) give the same values of S α , T α for eachpair ( x , w ) , ( x , W ) of the eight sets of separating variables obtained above. Conjugate momenta.
As was shown in a preliminary version of this paper [17], the separationcoordinates x and x commute with respect to both Poisson brackets on ˆ M defined by the Poissontensors ˆ P and ˆ Q . This noticeable property brings our separation procedure into the framework of thestandard Hamilton–Jacobi theory. (Curiously, after more than a century it is still unknown whetherthe same commutativity property is shared by the separating variables constructed by K¨otter in [5]).Without outlining here the transition from the algebraic geometric picture developed so far to theHamilton-Jacobi picture of Classical Mechanics, we simply recall that for x , x , the conjugate momentawith respect to the Poisson tensor ˆ P are y = M p Φ( x ) , y = L p Φ( x ) . So the algebraic coordinates w and W are related to the conjugate momenta according to w =2 p Φ( x ) y and W = − ˆ C √ y . The equations of the curves C and K become then the equations givingthe Hamiltonians H p and H s as functions of the separating coordinates x , x , and of their conjugatemomenta y , y . From these equations one may fully reconstruct the description of the Clebsch systemfrom the viewpoint of the classical Hamilton-Jacobi theory. We believe that the above exposition brings two new ideas in the field of integrable systems.The first one is the concept of separable Poisson pencils. They provide both an integrable Hamil-tonian system and the coordinates which allow to solve it by separation of variables. Such pencils arecharacterized by two properties: • Their polynomial Casimir functions satisfy the integrability condition m = n + 2 p ; • In the phase space there exist vector fields Z a (their number equals the number of polynomialCasimir functions) which solve a set of cohomological conditions of the type Z a ∧ d P Z a = 0 ( andsimilar, but more complicated, conditions with respect to the second Poisson tensor Q ).19his paper gives just a first example of this occurrence. For the Clebsch system the pencil has a singlepolynomial Casimir function of degree two. Nevertheless, the tools used in the paper allow extensionsto pencils with an arbitrary number of polynomial Casimir functions. Naturally, in this case the co-homological conditions increase in number and become more complicated. For instance, if the pencilhas two polynomial Casimir functions of degrees 1 and 2, there are 12 such conditions. However, theidea is always the same: the cohomological conditions imply the Kowalewski separability conditions,and therefore guarantee the existence of a set of separating coordinates for the integrable bihamiltoniansystems defined by the polynomial Casimir functions.The second idea is the lifting. Several examples show that in order to be able to solve the previouscohomological conditions one must accept the idea of constraining the dynamical vector fields to suitableinvariant submanifolds. From our standpoint, this is the correct interpretation of the pioneering workof Weber [15]. The lift counterbalances the need for a constraint. The process of lifting is a topic tobe still thoroughly investigated, yet a point seems to emerge clearly from the few examples we havestudied so far. There is a class of integrable and separable bihamiltonian systems for which there aretwo distinct algebraic curves supporting the separating coordinates. In other words, the Clebsch systemis not an isolated example. For this reason we believe that the study of the problem of inversion ofthe quadratures involving two different algebraic curves, which is considered in the second part of thepaper, deserves special attention. Acknowledgments
The authors acknowledge the support and hospitality of Department of Mathematics of Univer-sit´a di Milano Bicocca, where a part of this work has been made. The contribution of Y.F was par-tially supported by the Spanish MINECO-FEDER Grant MTM2015-65715-P and the Catalan grant2017SGR1049. T. Skrypnyk is grateful to the late B. Dubrovin for support and discussions.
Appendix : Proof of Proposition 4
On the seven-dimensional manifold ˆ M , the 3-vector ˆ Z ∧ d ˆ P ˆ Z has 35 components. They are computedby inserting any triple of coordinate functions ( J, K, L ) into the equation( Z ∧ d ˆ P Z )( dJ, dK, dL ) = X cyclic Z ( J ) ( Z { K, L } P − { Z ( K ) , L } P − { K, Z ( L ) } P ) , where the sum is over the cyclic permutations of the coordinate functions. Fifteen components vanishdue to the assumption that ˆ C is a Casimir function and that ˆ Z ( ˆ C ) = 0 ( just set L = ˆ C in theprevious equation). Hence, the problem is to study the vanishing of the remaining 20 components, bothin the case of the Poisson tensor ˆ P and in the case of the Poisson tensor ˆ Q . Since the unknown functions A α ( ζ ) and B α ( ζ ) do not depend on the coordinates ( S α , T α ), each of the above 40 equations split, byseparation of variables, into 3 ODE’s. So, one has to study the solvability of a system of 120 ODE’sin 6 unknown functions. This study is split in four Lemmas. For brevity, we shall agree to denote by aprime the derivative with respect to the coordinate ζ of any function defined on the curve E .20 emma 6 If the functions A α ( ζ ) and B α ( ζ ) verify the algebraic constraints X α =1 A α B α = 0 , X α =1 B α = 0 , and the differential equations B β X α =1 A α A ′ α − B ′ β X α =1 A α − A β X α =1 B α A ′ α = 0 , the 3-vector ˆ Z ∧ d ˆ P ˆ Z vanishes on ˆ S .Proof. Let us choose J = T , K = T , L = T . By taking into account the constraint ˆ C = 0, one readilyfinds that this component vanishes if and only if the functions B α ( ζ ) satisfy the differential equations B β X α =1 B α B ′ α − B ′ β X α =1 B α = 0 . In the same way, one finds that the component J = S , K = S , L = S vanishes if and only if thedifferential equations written in the Lemma are satisfied. To control the remaining components, letus preliminarily notice that the algebraic constraints listed in the Lemma imply the existence of amultiplier ρ such that ρB α = B β A γ − B γ A β . This multiplier is one of the roots of the quadratic equation ρ + P α A α = 0 since, otherwise, theabove linear system would have only the trivial solution B α = 0. By plugging the last relation into thedifferential equations one readily put them in the form ρB ′ α = B β A ′ γ − B γ A ′ β . They allow to check that the remaining 18 components of the 3-vector ˆ Z ∧ d ˆ P ˆ Z vanish on the surfaceˆ C = 0, without any further condition. (cid:3) Lemma 7
Assume that the seven functions A α ( ζ ) , B α ( ζ ) and v ( ζ ) satisfy the ten algebraic constraints A α + ( v + j β ) B γ + ( v + j γ ) B β = 0 , ( v + j α ) A α B β − ( v + j β ) A β B α = 0 ,A α A β − ( v + j γ ) B α B β = 0 , X α =1 ( v + j α ) A α = 0 . Then the 3-vector ˆ Z ∧ d ( ˆ Q + v ( ζ ) ˆ P ) ˆ Z vanishes on ˆ S . roof. One has to repeat the computation done before by replacing the Poisson tensor P with thePoisson pencil ˆ Q + v ˆ P . One finds that the component relative to the choice J = T , K = T , L = T vanishes, on the surface ˆ C = 0, if and only if the following system of differential equations B ′ α ( A α + ( v + j β ) B γ + ( v + j γ ) B β )++( A α A β − ( v + j γ ) B α B β ) B ′ β + ( A α A γ − ( v + j γ ) B α B γ ) B ′ γ = 0is satisfied. Similarly, one recognizes that the choice J = S , K = S , L = S leads to the differentialequations B ′ α X α =1 ( v + j α ) A α + (( v + j α ) A α B β − ( v + j β ) A β B α ) A ′ β ++(( v + j α ) A α B γ − ( v + j γ ) A γ B α ) A ′ γ = 0 . One may notice that the coefficients of these differential equations are the quadratic polynomials listedin Lemma 7. So the algebraic constraints imposed by the Lemma define a singular solution of thesedifferential equations.The inspection of the remaining 18 components confirms this occurrence in general. (cid:3)
Since the vector field ˆ Z does not affect the coordinate ζ , to say that the 3-vectors ˆ Z ∧ d ˆ P ˆ Z andˆ Z ∧ d ( ˆ Q + v ( ζ ) ˆ P ) ˆ Z vanish is the same as to say the the 3-vectors ˆ Z ∧ d ˆ P ˆ Z and ˆ Z ∧ d ˆ Q ˆ Z separately vanish.So the first two Lemmas provide an overdetermined set of conditions entailing that the vector field ˆ Z satisfies the Frobenius conditions with respect to the Poisson pencil ˆ Q − λ ˆ P on the submanifold ˆ S . Thenext two Lemmas show how to reduce this overdetermined system of conditions. Lemma 8
The twelve algebraic constraints imposed by the first two Lemmas are equivalent to the fourquadratic equations (5) presented in the Introduction.Proof.
The first two Eq.s (5) are already contained in the set of 12 algebraic constraints. The thirdequation follows from the obvious identity X α =1 A α + ( v + j β ) B γ + ( v + j γ ) B β = X α =1 (( j β + j γ ) B α + A α ) + 2 v X α =1 B α . The fourth equation is proved by noticing that the 12 conditions imply X α =1 ( v + j β )( v + j γ ) B α = 0 . By expanding this identity in powers of v , and keeping in mind that v = − P α =1 j α A α P α =1 A α v according to v = − P α =1 j α A α P α =1 A α in such a way to satisfy the last of the 12 conditions. Then one has to keep in mind the parametricrepresentations A α = c α v β v γ , B α = c α v α of the functions A α ( ζ ) and B α ( ζ ) introduced in Section 3, and the relation v α = v + j α . With thesepositions, the 12 algebraic constraints follow immediately. (cid:3) Lemma 9
The six elliptic functions defined by the four quadratic constraints (5) solve the three residualODE’s imposed by Lemma 6Proof.
Let us denote, for brevity, the left-hand side of the differential equations by the symbol Eq ( α ), andlet us notice that the four quadrics imply the further identity P α =1 j α A α B α = 0. Then one easily sees that:1) The expression P α =1 A α Eq ( α ) vanishes on account of the constraint P α =1 A α B α = 0; 2) The expression P α =1 B α Eq ( α ) vanishes on account of the constraints P α =1 A α B α = 0 and P α =1 B α = 0; 3) The expression P α =1 j α B α Eq ( α ) vanishes on account of the constraints P α =1 j α A α B α = 0 and P α =1 ( j β + j γ ) B α + A α = 0. Sincethree independent linear combinations of the differential equations vanish on account of the algebraicconstraints, one conclude that the equations themselves vanish. This Lemma concludes the proof ofProposition 4. (cid:3) References [1] A. Clebsch. ¨Uber die Bewegung eines K¨orpers in einer Fl¨ussigkeit.
Math. Ann. , 1871, (3) , 238-261.[2] Schottky F., ¨Uber das analytische Problem der Rotation eines starren K¨orpers in Raume von vierDimensionen,
Sitzungsber., K¨onig. Preuss. Akad. Wiss., Berlin, 12 (1891), 227–232.[3] Adler M., van Moerbeke P. and Vanhaecke P.,
Algebraic integrability, Painleve geometry and Liealgebras. , Springer, (2013). 234] Adler M., van Moerbeke P.,
The algebraic integrability of the geodesic flow on SO (4) , InventionesMathematicae, (1982).[5] K¨otter F.,
Uber die Bewegung eines festen K¨orpers in einerFl¨ussigkeit. I, II,
J. reine angew. Math.,109 (1892), 51–81, 89–111.[6] Belokolos E.D., Bobenko A.I., Enol’sii V.Z., Its A.R., and Matveev V.B. ,
Algebro-GeometricApproach to Nonlinear Integrable Equations,
Springer Series in Nonlinear Dynamics, Springer-Verlag,1994.[7] Zhivkov. A, Christov, O.,
Effective solutions of Clebsch and C. Neumann systems. edoc.hu-berlin.de, (1998).[8] Sklyanin E. K.,
Separation of variables: New trends , Progr. Theor. Phys., 118, (1995), 35-60.[9] Adler M., van Moerbeke P. ,
Geodesic flow on so (4) and the intersections of quadrics, Proc.Natl.Acad. Sci. USA, 81, (1984), 4613–4616.[10] Haine L.,
Geodesic flow on so (4) and Abelian surfaces, Math. Ann., 263 (1983), 435–472.[11] Magri F.,
The Kowalevski’s top and the method of syzygies , Ann. Inst. Fourier, 55, (2005), 2146-2159.[12] Magri F.,
The Kowalevski top revisited , in Vol. 1 of Integrable sytems and algebraic geometry, R.Donagi, T. Shaska ed.s, London Mathematical Society Lecture Notes Series , (2020), 329-355.[13] Magri F.,
The Kowalevski separability conditions , in Dubrovin Memorial volume, I. Krichever, S.Novikov, O. Ogievetsky, S. Shlomann ed.s, Proceedings of Symposia in Pure Mathematics bookseries, AMS (to appear).[14] Pantazis S.,
Prym varieties and the geodesic flow on SO ( n ) , Math. Ann., (1986) 273–297.[15] Weber H.,
Anwendung der Thetafunctionen zweir Veranderlicher auf die Theorie der Bewegungeines festen K¨orpers in einer Fl¨ussigkeit’,
Math. Ann., 14 (1878), 173–206.[16] Gel’fand I.M., Zakharevich I.,
On the Local Geomatry of Bihamiltonian Structures,
The Gel’fandMathematical Seminars 1990-1992 ,Birkhauser, Boston, 1993.[17] Magri F., Skrypnyk T.,