Symmetry, Integrability and Geometry: Methods and Applications New Variables of Separation for the Steklov–Lyapunov System

A rigid body in an ideal fluid is an important example of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(3) = so(3)\ltimes\mathbb R^3$. We present the bi-Hamiltonian structure and the corresponding variables of separation on this phase space for the Steklov-Lyapunov system and it's gyrostatic deformation.


Introduction
Rigid body dynamics in an ideal incompressible fluid is rich with problems interesting from mathematical point of view, in particular, the research of integrable problems. Certainly, the most famous are the three integrable cases under the names of Kirchhoff, Clebsch, Steklov and Lyapunov. The latter two cases are more interesting because there are no a obvious symmetry groups associated with the additional integrals of motion.
All these classical cases were discovered and carefully studied in the 18th and 19th centuries [5]. For instance, the Kirchhoff equations for the Clebsch and Steklov-Lyapunov cases were first solved explicitly by Kötter after some mysterious separation of variables [11,12]. At the moment no separation which is alternative to his original separation of variables is known for these systems, even though there is a lot of literature dedicated to the problem, including the theta-functions solutions, associated with the Lax matrices, and the detailed geometric descriptions of the invariant surfaces on which the motions evolve, see books [1,2] and references within.
In this paper we apply the bi-Hamiltonian geometry to the direct calculation of variables of new separation for the Steklov-Lyapunov system and it's Rubanovsky generalization. We have to notice right away, that our main purpose is the development of the bi-Hamiltonian geometry instead of integration of particular equations of motion using separation of variables method.
An integrable system is separable, if there are n separation relations Φ i (u i , p u i , H 1 , . . . , H n ) = 0, i = 1, . . . , n, with det ∂Φ i ∂H j = 0, (1.1) connecting single pairs (u i , p u i ) of canonical variables of separation with the n functionally independent Hamiltonians H 1 , . . . , H n . Solving these relations in terms of p u i one gets the Jacobi equations and the corresponding additively separable complete integral of the Hamilton-Jacobi equation Of course, any variables of separation are determined up to the trivial transformation (1.2) which preserve all the properties of algebraic curves defined by separated relations (1.1). However, if we have another separated relations for the same integrals of motion . . , H n ) = 0, i = 1, . . . , n, with det ∂Ψ i ∂H j = 0, which can not be reduced to initial ones (1.1) by trivial change of variables (1.2), we usually say about different variables of separation. Of course, any two families of canonical variables on a given phase space are related by a generic canonical transformations v i = g i (u 1 , . . . , u n , p 1 , . . . , p n ),  [18,22,24], as well as in distinct procedures of quantization and various methods of qualitative analysis, etc.
The milestones of the variables separation technique include the works of Stäckel, Levi-Civita, Eisenhart, Benenti and others. The majority of results was obtained for a very special class of integrable systems, important from the physical point of view, namely for the natural Hamiltonian systems with quadratic in momenta integrals of motion on cotangent bundles to Riemannian manifolds. The Kowalevski and Chaplygin results on separation of variables for the systems with higher-order integrals of motion have been missed out of this scheme until recently [18,22,24].
In the Steklov-Lyapunov case we have quadratic integrals of motions, but the phase space is the Poisson manifold instead of the cotangent bundle to Riemannian manifold. So, in this case we can use neither the Levi-Civita criteria, nor the Eisenhart-Benenti theory. Below we show how variables of separation for the given integrable system may be calculated without any additional information (Killing tensors, Lax matrices, r-matrices, links with soliton equations etc.). velocities of the body. All these vectors in R 3 are expressed in the body frame attached to the body originating at the center of buoyancy [9].
A rigid body in the ideal fluid is an important example of Hamiltonian systems on a dual to Lie algebra e(3) = so(3) R 3 . The dual space e * (3) is the Poisson manifold endowed with the canonical Lie-Poisson brackets where ε ijk is a totally skew-symmetric tensor. There are two Casimir elements where x, y means scalar product of two three-dimensional vectors x, y ∈ R 3 . As usual [9], element M ∈ so(3) is identified with three-dimensional vector M ∈ R 3 using well known isomorphism of the Lie algebras (R 3 , ×) and so (3), where × is a cross product, [·, ·] is a matrix commutator and index µ means a 3×3 antisymmetric matrix associated with the vector z. Using this agreement we can rewrite canonical Poisson bivector on e * (3) in the following compact form Here, symmetry means that the Hamiltonian that describes the dynamics in T * E(3) is an invariant to actions of E(3), i.e., one can translate the inertial frame or rotate it in any direction without affecting the equations of motion [15].
The Steklov-Lyapunov case of the rigid body motion is characterized by the following secondorder homogeneous polynomial integrals of motion where wedge denotes an adjoint matrix, i.e. a cofactor matrix A ∨ = (det A)A −1 . For integrability A has to be symmetric matrix, which may be reduced to the diagonal form

Separation of variables by Kötter
The explicit integration of the classical Steklov-Lyapunov systems via separation of variables had been first made by F. Kötter in 1900 [11]. Here we want to add some new details in the known coincidence of the Kötter variables of separation v 1,2 with the elliptic coordinates on the sphere. According to [23,25] there is a Poisson map, which identifies the Steklov-Lyapunov system with a system that describes motion on the surface of a unit two-dimensional sphere S 2 in a fourth-degree polynomial potential field. This dynamical system is separable in standard elliptic coordinates on the sphere, and the inverse Poisson map allows us to get complete solution of the Steklov-Lyapunov system.
then the Poisson map f : (p, M ) → (x, J), defined by relates manifold e * (3) with coordinates (p, M ) and cotangent bundle T * S 2 to the unit twodimensional sphere S 2 with coordinates (x, J).
The proof consists in the verification of the Lie-Poisson brackets between variables x and J and calculation of the corresponding Casimir functions x, x = |x| 2 = 1, and x, J = 0.
So, this coajoint orbit of e * (3) with coordinates x and J is simplectomorphic to cotangent bundle T * S 2 to the unit two-dimensional sphere S 2 , see for instance [15].
Inverse Poisson map f −1 : (x, J) → (p, M ) looks like where functions α, β on x, J are solutions of the following equations with the natural Hamilton function on T * S 2 Here C 1,2 are values of the Casimir functions (2.3).
The proof of this proposition and all the details may be found in [23,25].
Remark 3. It is well-known that the generic level sets of the Casimir functions (coadjoint orbits) on e * (3) are only diffeomorphic to the cotangent bundle T * S 2 . It allows us to directly connect the Kirchhoff equations with the equations of motion by geodesic on S 2 , however, at C 2 = 0 we have to destroy the standard symplectic structure on T * S 2 by adding some "monopole" terms [15].
In the Steklov-Lyapunov case we use the Poisson map (2.9), which preserves the standard symplectic structure on T * S 2 . As a punishment for this preservation of the standard symplectic structure we have to consider potential motion on S 2 instead of the geodesic motion.
The Hamiltonians H 1,2 (x, J) (2.11), (2.12) on T * S 2 are separable in the elliptic (spheroconical) coordinates v 1,2 , which are zeroes of the function This variables of separation satisfy the following separated relations where the three-dimensional vector (λ) is the so-called Kötter vector with entries depending on the auxiliary variable λ (spectral parameter). The explicit solution of the corresponding Abel-Jacobi equations in theta-functions was given in [11]. In order get initial variables p, M as functions on time variable we have to substitute solutions of the Abel-Jacobi equations into the variables on T * S 2 where (i, j, k) is permutation of (1, 2, 3), and then into the variables p and M (2.10). Another modern verification of the Kötters calculations may be found in [4,8].

Calculation of the variables of separation in bi-Hamiltonian geometry
We can only guess how Kötter invented the variables of separation v 1,2 , which coincide with the elliptic coordinates on the auxiliary two-dimensional sphere, because he gave no explanations of calculations in very brief communication [11]. It is clear that behind the striking formulas there must be a certain geometric idea, but the domain of applicability of this idea is usually restricted by a partial model under consideration. For instance, we can not apply the Kötter separation to the Kowalevski top and Kowalevski separation to the Steklov-Lyapunov system etc.
Our aim is to discuss some algorithm of calculation of the variables of separation in the framework of the bi-Hamiltonian geometry, which is applicable to many known integrable systems [7,18,20,21,22,24,26]. In fact this algorithm consists of the following steps: • calculate the second Poisson bracket compatible with canonical one starting with the given integrals of motion in the involution with respect to this canonical Poisson bracket; • if the Poisson brackets have different symplectic leaves, calculate a projection of the second bracket on symplectic leaves of the first bracket; • calculate coordinates of separation as eigenvalues of the corresponding recursion operator; • calculate the canonically conjugated momenta with respect to the first Poisson bracket; • calculate the separated relations.
The input of algorithm is a set of integrals of motion and canonical Poisson bracket, whereas output is a set of separated relations. All details about construction of a suitable projection are discussed in [7]. Because variables of separation (1.1) are defined up to canonical transformations u i → f (u i , p u i ) on the first step of this algorithm we have to narrow the search space using some artificial tricks. It is a main technical problem of this method. The second technical problem is the calculation of the momenta conjugated to obtained coordinates, see [7,18,22,24].

Polynomial and rational Poisson brackets on e * (3)
Bi-Hamiltonian structures can be seen as a dual formulation of integrability and separability, in the sense that they substitute a hierarchy of compatible Poisson structures to the hierarchy of functions in involution, which may be treated either as integrals of motion or as variables of separation. So, our first step is calculation of the second Poisson bivector P compatible with kinematic Poisson bivector P .
According to [20,26] any separable system is a bi-integrable system, i.e. integrals of motion H k (1.1) are in bi-involution  [20,26]. For the given integrable system fixed by a kinematic bivector P and a tuple of integrals of motion H 1 , . . . , H n bi-Hamiltonian construction of variables of separation consists in a direct solution of the equations (3.1) and (3.2) with respect to an unknown bivector P . The main problem is that the geometrically invariant equations (3.1), (3.2) have a'priory infinite number of solutions [18,20,21,22,24,26].
In order to get a search algorithm of effectively computable solutions we have to narrow the search space by using some non-invariant additional assumptions. According to [20,21] hereafter we assume that P has the same foliations by symplectic leaves as P , i.e. that P dC 1,2 = 0 (3.3) and P doesn't have any other Casimir elements. The geometric meaning of this restriction is discussed in [20,21]. In fact it allows us to avoid calculations of the projection of the second bracket on the symplectic leaves of the first bracket.
In the Steklov-Lyapunov case solving equations (3.1), (3.2) and (3.3) in the space of homogeneous second-order polynomial bivectors and of rational bivectors with the second-order homogeneous numerators and linear denominators we obtain the following two propositions.
As above × is a cross product, the antisymmetric matrix z µ is defined by vector z (2.4) and the matrix (x ⊗ y) ij = x i y j is determined by a pair of vectors x and y. In order to explain this notations we write out the corresponding Poisson brackets The second brackets are equal to We have to stress that this brackets are defined over complex field because c, c = c 2 1 +c 2 2 +c 2 3 = 0.
The proof is a straightforward calculation.
Remark 4. In fact, polynomial bivector P 1 has been obtained in [21] as an incidental result by investigation of the Poisson bivectors on the Lie algebra so * (4) and the corresponding integrable cases in the Euler equations on so * (4). Now we recover this bivector by solving equations (3.1), (3.2) and (3.3) for the Steklov-Lyapunov system.
Let us briefly discuss the bi-Hamiltonian structure related with the Kötter variables of separation. Elliptic coordinates on the sphere S 2 (2.13) are associated with the polynomial Poisson bivector where L X is a Lie derivative along the vector field X = X j ∂ j with the following entries: Bivector P e is compatible with P and has the same foliation by symplectic leaves as P .
Using the Poisson map (2.9), (2.10) we can easily express P ell in the initial variables p, M . It will be a rational bivector P ell = R/Q, where R is a bivector with fourth-order homogeneous polynomial entries and Q = |(M − Bp) × p| 2 is a fourth-order polynomial as well. So, we could directly calculate the Kötter variables solving equations (3.1), (3.2) and (3.3) in the corresponding space of rational bivectors.

Calculation of variables of separation
The bi-involutivity of the integrals of motion (3.1) is equivalent to the existence of control matrix F defined by The additional assumption (3.3) ensures that F is a non-degenerate matrix and the eigenvalues of F are the desired variables of separation [20,21]. Moreover, for the so-called Stäckel separable systems the suitable normalized left eigenvectors of the control matrix F form the Stäckel matrix S [20,21,22,24]. In this case separated relations (1.1) are affine equations in integrals of motion H k . Let us calculate the control matrices for the Steklov-Lyapunov system and, for brevity, introduce three constants and some linear functions on variables p, M ρ k = c, A k p , σ k = c, A k M , k = 0, 1, 2, (3.9) which are related to each other via the Casimir functions (2.3) on e * (3). For instance, In this notations control matrices associated with the Hamiltonians H 1,2 (2.7) and the Poisson bivectors P 1,2 (3.4), (3.5) look like (3.11) Now we can simply calculate the desired variables of separation using two control matrices. Namely, let u 1,2 be eigenvalues of the control matrix F 2 whereas the eigenvalues of F 1 be doubled momenta 2p u 1,2 , so that the characteristic polynomial has the form Another equivalent definition of momenta p u i is given by a relation Now we can prove the following Proposition 5. On symplectic leaves of e * (3) variables u 1,2 and p u 1,2 are canonical variables with respect to canonical Poisson bracket (2.2).
The proof consists of the calculation of the Poisson brackets between coefficients of characteristic polynomials A(λ) (3.13) and B(λ) (3.12). Now we have to determine an inverse transformation from variables u 1,2 and p u 1,2 to initial variables p, M . Firstly, using the definitions (3.13), (3.12) and the relation (3.10), we express five linear functions (3.9) via variables of separation and Casimir functions , 14) It is easy to see that ρ k and σ k are symmetric functions in u 1,2 , p u 1,2 .
Secondly, we determine the initial variables p, M as functions on ρ k and σ k : Here c i are entries of the vector c (3.8), C 1,2 are the Casimir functions on e * (3) (2.3) and (i, j, k) means the permutation of (1, 2, 3). Matrices F 1,2 (3.11) in canonical variables of separation look like where the Stäckel matrix S is equal to The notion of the Stäckel matrix S allows us to easily get the separated relations (1.1) and prove that canonical variables u, p u are the variables of separation for the Steklov-Lyapunov system.

Proposition 6.
In the Steklov-Lyapunov case the canonical variables u 1,2 (3.12) and p u 1,2 (3.13) satisfy the following separated relations where cubic polynomials ϕ 3 (u) and φ 3 (u) are equal to The proof consists of substituting integrals of motion H 1,2 (2.7) in terms of variables of separation (3.15) into the separated relations (3.17).
So, in the Steklov-Lyapunov case equations of motion are linearized on Jacobian of the genus two hyperelliptic curve defined by the equation Φ(u, p u ) = 0 (3.17) and the system of the Abel-Jacobi equations has the standard form Here p(u) means the function p u on u obtained from the equation (3.17), β 1,2 are certain constants depending only on the choice of the Hamiltonian (H 1 or H 2 ) and γ 1,2 are two constants. Solving these equations with respect to u 1,2 (t, γ 1,2 ) and substituting these solutions into the expressions (3.14) and (3.15) we finally get the initial variables p, M as functions on time variable t and six constants H 1,2 , C 1,2 and γ 1,2 .
Remark 5. In order to give an explicit theta-functions solution, one can apply the standard machinery of the Weierstrass root functions describing inversion of the hyperelliptic quadratures (3.19), completely similar to solution of Jacobi's geodesic problem or Neumann's particular case of the Clebsch system [5,8,27].
Using shift of coordinates we can rewrite separated relations (3.17) in the following form Here P , Q, R, T are the sixth-order polynomials in momenta, Q, T are the sixth-order polynomials in coordinates, whereas P and R are the seventh-and eighth-order polynomials in coordinates. We did not find a foreseeable expressions for these polynomials or their combinations. In any case variables u, p u and v, p v are related by non-trivial canonical transformation (1.3).
Remark 6. The Steklov-Lyapunov system on e * (3) coincides with the Steklov system on so * (4) after some linear change of phase variables [3]. It is a twisted Poisson map, which permutes first and second Lie-Poisson brackets on e * (3) and so * (4) [19]. We suppose that variables u 1,2 and p u 1,2 coincide with the complex variables of separation for the Steklov system on so(4) introduced in [6,13] up to this change of variables and transformation of the canonical momenta associated with permutation of the Poisson brackets. We thank one of the referees for the reference on these papers. Algebro-geometric relations of this complex coordinates u 1,2 with the real Kötter coordinates v 1,2 is discussed in [8].

The Rubanovsky system
Let us consider a nontrivial integrable generalization of the Steklov-Lyapunov system discovered by Rubanovsky [16] where H 1,2 are given by (2.7) and b = (b 1 , b 2 , b 3 ) is a constant vector. This deformation describes the motion of a gyrostat in an ideal fluid under the action of the Archimedes torque, which arises when the barycenter of the gyrostat does not coincide with its volume center. The problem of separation of variables for the Rubanovsky systems was unsolved up until now. The Rubanovsky integrals of motion are non-homogeneous second-order polynomials and, therefore, it is natural to solve the equations (3.1) and (3.2) in the space of non-homogeneous second-order polynomial bivectors and in the similar space of rational bivectors. and one function is different This shift ofσ 1 acts only on M variables (3.15).
Proposition 8. In the Rubanovski case the separated relations have the following form Here cubic polynomials ϕ 3 and φ 3 are given by (3.18) and D = tr A − 3A.
The proof consists of substituting integrals of motionĤ 1,2 in terms of variables of separationû,p u into the separated relations (3.24).
As above, we can prove that the equations of motion are linearized on the Jacobian variety of the genus two hyperelliptic curve defined by (3.24). For the brevity, here we omit the explicit expressions for a base of holomorphic differentials and the corresponding Abel-Jacobi equations, which may be easily obtained using modern computer algebra software.
Calculation of the separating variables and the corresponding algebraic curve for the Rubanovski gyrostat is a new result, which allows us to make conclusion an applicability of bi-Hamiltonian methods to study finite-dimensional integrable Hamiltonian systems.