From su(2) Gaudin Models to Integrable Tops

In the present paper we derive two well-known integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) performing an algebraic contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin models. The procedure preserves the linear r-matrix formulation of the ancestor models. We give the Lax representation of the resulting integrable systems in terms of su(2) Lax matrices with rational and elliptic dependencies on the spectral parameter. We finally give some results about the many-body extensions of the constructed systems.


Introduction
The Gaudin models were introduced in 1976 by M. Gaudin [5] and attracted considerable interest among theoretical and mathematical physicists, playing a distinguished role in the realm of integrable systems. Their peculiar properties, holding both at the classical and at the quantum level, are deeply connected with the long-range nature of the interaction described by its commuting Hamiltonians, which in fact yields a typical "mean field" dynamics.
Indeed the Gaudin models describe completely integrable classical and quantum long-range spin chains. The original Gaudin model was formulated as a quantum spin model related to the Lie algebra su(2) [5]. Later it was realized that such models can be associated with any semisimple complex Lie algebra g [6,10] and a solution of the corresponding classical Yang-Baxter equation [2,23]. An important feature of Gaudin models is that they can be formulated in the framework of the r-matrix approach. In particular, they admit a linear r-matrix structure that characterizes both the classical and the quantum models, and holds whatever be the dependence (rational (XXX), trigonometric (XXZ), elliptic (XYZ)) on the spectral parameter. In this context, it is possible to see Gaudin models as appropriate "semiclassical" limits of the integrable Heisenberg magnets [26], which admit a quadratic r-matrix structure.
In the 80's, the rational Gaudin model was studied by Sklyanin [24] and Jurčo [10] from the point of view of the quantum inverse scattering method. Precisely, Sklyanin studied the su(2) rational Gaudin models, diagonalizing the commuting Hamiltonians by means of separation of variables and stressing the connection between his procedure and the functional Bethe Ansatz.
On the other hand, the algebraic structure encoded in the linear r-matrix algebra allowed Jurčo to use the algebraic Bethe Ansatz to simultaneously diagonalize the set of commuting Hamiltonians in all cases when g is a semi-simple Lie algebra. We have to mention here also the the work of Reyman and Semenov-Tian-Shansky [22]. Classical Hamiltonian systems associated with Lax matrices of the Gaudin-type were studied by them in the context of a general group-theoretic approach.
Vadim Kuznetsov, to whom this work is dedicated, widely studied Gaudin models, especially from the point of view of their separability properties [11,12,13] and of their integrable discretizations through Bäcklund transformations [8,14]. In [14] we collaborated with him showing that the Lagrange top can be obtained through an algebraic contraction procedure performed on the two-body su(2) rational Gaudin model. Such a derivation of the Lagrange system preserves the linear r-matrix algebra of the ancestor model, and it has been used as a tool to construct an integrable discretization starting from a known one for the rational su(2) Gaudin model [8].
The purpose of the present paper is twofold: on one hand we recall the procedure we used in [14] to obtain the Lagrange top from the two-body su(2) rational Gaudin model; on the other hand we show how the same technique can be used to derive a special case of the Clebsch system (i.e. the motion of a free rigid body in an ideal incompressible fluid) starting from the elliptic su(2) Gaudin model. In the last Section we show how to construct many-body extensions starting from the obtained Lax matrices governing the Lagrange top and the Clebsch system.

A short review of su(2) Gaudin models
The aim of this Section is to give a terse survey of the main features of su(2) Gaudin models. In particular we shall describe them in terms of their (linear) r-matrix formulation, providing their Lax matrices and r-matrices. For further details we remand at the references [5,6,8,10,11,17,20,22,24,25,26].
Let us choose the following basis of the linear space su(2): We recall that the correspondence is an isomorphism between (su(2), [ ·, · ]) and the Lie algebra (R 3 , ×), where × stands for the vector product. This allows us to identify R 3 vectors and su(2) matrices. We supply su(2) with the scalar product ·, · induced from R 3 , namely a, b = −2 tr (a b) = 2 tr (b a † ), ∀ a, b ∈ su(2). This scalar product allows us to identify the dual space su * (2) with su(2), so that the coadjoint action of the algebra becomes the usual Lie bracket with minus. The Lie-Poisson algebra of the N -body su(2) Gaudin models is given by (minus) ⊕ N su * (2). We will denote by {y α i } 3 α=1 , 1 ≤ i ≤ N , the set of the (time-dependent) coordinate functions relative to the i-th copy of su (2). Consequently, the Lie-Poisson brackets on ⊕ N su * (2) read Here ǫ αβγ is the skew-symmetric tensor with ǫ 123 = 1. The brackets (2.1) are degenerate: they possess the N Casimir functions that provide a trivial dynamics.
The su(2) rational, trigonometric and elliptic Gaudin models are governed respectively by the following Lax matrices defined on the loop algebra su(2)[ λ, λ −1 ]: where the λ i 's, with λ i = λ k , 1 ≤ i, k ≤ N , are complex parameters of the model. We remark that in equation (2.5) cn(λ), dn(λ), sn(λ) are the elliptic Jacobi functions of modulus k. In equation (2.3) p is a constant vector in R 3 . Its presence is necessary in the rational case in order to get a sufficient number of functionally independent integrals of motion. It is well-known that the Lax matrices (2.3), (2.4) and (2.5) describe completely integrable systems on the Lie-Poisson manifold associated with ⊕ N su * (2). In particular they admit a linear r-matrix formulation, which ensures that all the spectral invariants of L r G (λ), L t G (λ), L e G (λ) form a family of involutive functions. Let us give the following result.
In equation (2.6) 1 denotes the 2 × 2 identity matrix and ⊗ stands for the tensor product in In the rational case the r-matrix is equivalent to r r (λ) = −Π/(2 λ), where Π is the permutation operator in C 2 ⊗ C 2 .

first integrals of motion and the integrals
are the Casimir functions given in equation (2.2).
, provides a set of 2N independent involutive integrals of motion given by are the Casimir functions given in equation (2.2).

provides a set of 2N independent involutive integrals of motion given by
, Here θ αβ (λ), α, β = 0, 1, is the theta function 1 , and θ αβ . In the rational case it is possible to select a simple and remarkable Hamiltonian. It is given by the following linear combination of the integrals of motion {H r i } N i=1 given in equation (2.8):

9)
1 We are using the notation adopted in [26]: where the η i 's with η i = η k , 1 ≤ i, k ≤ N , are arbitrary complex numbers. An interesting specialization of the Hamiltonian (2.9) is obtained considering Proposition 5. The equations of motion w.r.t. the Hamiltonian (2.10) are given bẏ ) admit the following Lax representation: with the matrix L r G (λ) given in equation (2.3) and

A Lagrange top arising from the rational su(2) Gaudin model
Recall that the (3-dimensional) Lagrange case of the rigid body motion around a fixed point in a homogeneous field is characterized by the following data: the inertia tensor is given by diag (1, 1, α), α ∈ R, which means that the body is rotationally symmetric with respect to the third coordinate axis, and the fixed point lies on the symmetry axis [1,4,14,22]. As noticed in [14] the Lagrange top can be obtained from the two-body rational su(2) Gaudin model performing the contraction procedure previously described.
Let us recall the main features of the dynamics of the Lagrange top (in the rest frame). The equations of motion are given by: where m ∈ R 3 is the vector of kinetic momentum of the body, a ∈ R 3 is the vector pointing from the fixed point to the center of mass of the body and p . = (0, 0, p) is the constant vector along the external field. An external observer is mainly interested in the motion of the symmetry axis of the top on the surface a, a = constant. For an actual integration of this flow in terms of elliptic functions see [7].
A remarkable feature of the equations of motion (3.8) is that they do not depend explicitly on the anisotropy parameter α of the inertia tensor [4]. Moreover they are Hamiltonian equations with respect to the Lie-Poisson brackets of e * (3), see equation (3.2). The Hamiltonian function that generates the equations of motion (3.8) is given by Performing the same procedure on the Hamiltonian H r G .
= λ 1 H r 1 +λ 2 H r 2 given in equation (2.10) (with N = 2) and on the linear integral H r 1 + H r 2 = p, y 1 + y 2 we recover the integrals of motion of the Lagrange top. We have: = y 2 , y 2 /2 just Casimir functions. Hence, Finally, H r 1 + H r 2 = p, y 1 + y 2 = p, m = I r 2 . The same procedure allows one to recover the auxiliary matrices M (r,±) (λ) given in equation (

A Clebsch system arising from the elliptic su(2) Gaudin model
Let us now consider the Lax matrix given in equation (3.6) obtained performing the contraction procedure on the Lax matrix of the su(2) elliptic Gaudin model with N = 2.
A direct computation shows that the spectral invariants of L e (λ) are given by the following quadratic functions: Obviously, the choice k = 0 in the integrals (3.11) and (3.12) provides the spectral invariants of the trigonometric Lax matrix L t (λ) given in equation (3.5). Thus the system described by L t (λ) is a subcase of the one described by L e (λ). The quadratic functions (3.11) and (3.12) are in involution w.r.t. the Lie-Poisson brackets on e * (3) thanks to the r-matrix formulation in equation (3.7). Let us now recall the main features of the (3-dimensional) Clebsch case of the free rigid body motion (in an ideal fluid) [22,27]. This problem is traditionally described by a Hamiltonian system on e * (3) with the Hamiltonian function 13) where (m, a) ∈ e * (3) and the matrices A . = diag(α 1 , α 2 , α 3 ) and B . = diag(β 1 , β 2 , β 3 ) are such that the following relation holds: for some matrix C . = diag(γ 1 , γ 2 , γ 3 ). Taking into account equations (3.11)-(3.12) and (3.14) we see that C = diag(0, k 2 , k 2 −1) = B 1 for the Hamiltonian (3.11) and C = diag(1−k 2 , 1, 0) = A for the Hamiltonian (3.12). Thus L e (λ) can be considered as the Lax matrix of a special case of the Clebsch system described by the Hamilton function (3.13).
Remark 2. We note that the "traditional" Lax representations for the Hamiltonian flows (3.15)-(3.16) are given in terms of Lax matrices depending rationally on the spectral parameter [22,27]. However a Lax representation with elliptic dependence on the spectral parameter for the Clebsch system is already known [3]. Hence the novelty of our results consists just in establishing of the connection between su(2) elliptic Gaudin models and the Clebsch system.

Integrable chains of interacting tops
As shown in [17,19,20] one can construct integrable many-body systems starting with the onebody Lax matrices given in equations (3.4), (3.5) and (3.6). Such systems describe completely integrable (long-range) chains of interacting tops on the Lie-Poisson manifold associated with ⊕ M e * (3), being M the number of tops appearing in the chain. Moreover they admit the same linear r-matrix formulation given in equation (2.6) [17,20].
The above brackets are degenerate: they possess the following 2M Casimir functions: where the µ i 's with µ i = µ k , 1 ≤ i, k ≤ M , are complex parameters of the models. The Lax matrix L r M (λ) describes a system of M interacting Lagrange tops, called Lagrange chain in [17], while the matrices L t M (λ), L e M (λ) govern the dynamics of M interacting Clebsch systems. The latter models can be called Clebsch chains.
We now construct the spectral invariants of the Lagrange chain and of the Clebsch chain with k = 0.

The Lagrange chain
The complete set of integrals of the model can be obtained in the usual way. In fact, a straightforward computation leads to the following statement.

involutive first integrals of motion and the integrals {C
are the Casimir functions given in equation (4.1).
Notice that, as in the su(2) rational Gaudin model, there is a linear integral given by

Concluding remarks and open problems
In the present paper we have proposed an algebraic technique which enabled us to derive two (3dimensional) integrable cases of rigid body dynamics (the Lagrange top and the Clebsch system) from two-body su(2) Gaudin models. We remark that the explicit construction of the Lagrange top starting from the su(2) rational two-body Gaudin system has been presented for the first time in [14]. To the best of our knowledge the derivation of the Clebsch system defined by the involutive Hamiltonians (3.11)-(3.12) starting from the su(2) elliptic two-body Gaudin system is new, although the novelty is essentially in establishing of the connection between these two integrable systems.
Let us stress that the construction outlined here is just a top of an iceberg. In [17,18,19,20] we presented a general and systematic reduction, based on generalized Inönü-Wigner contractions, of classical Gaudin models associated with a simple Lie algebra g. Suitable algebraic and pole coalescence procedures performed on the N -pole Gaudin Lax matrices, enabled us to construct one-body and many-body hierarchies of integrable models sharing the same (linear) r-matrix structure of the ancestor models. This technique can be applied to any simple Lie algebra g and whatever be the dependence (rational, trigonometric, elliptic) on the spectral parameter. Fixing g = su(2), we constructed the so called su(2) hierarchies [18,20]. In particular the Lagrange top corresponds to the first element (N = 2) of the su(2) rational hierarchy, and the Clebsch system is the first element of the su(2) elliptic hierarchy.
We studied also the problem of discretizing the Hamiltonian flows of the su(2) rational Gaudin model. One of the authors (O.R.), together with Vadim Kuznetsov and Andy Hone, constructed in [8] one-point (complex) and two-point (real) Bäcklund transformations (BTs) for this model. Later on, in [14], again in collaboration with Vadim, we studied the problem of discretizing the dynamics of the Lagrange top using the BTs approach [15,16].
In [20,21], using a different approach, we have obtained a new integrable discretization for the Hamiltonian flow given in equation (2.11). It is expressed in terms of an explicit Poisson map and a suitable contraction performed on it enables us to construct discrete-time versions of the whole su(2) rational hierarchy. Our results include, as a special case (N = 2), the discretetime version of the Lagrange top found by Yu.B. Suris and A.I. Bobenko in [4]. Moreover, the same procedure enabled us to find an integrable discretization of the Hamiltonian flow (4.6), describing a discrete-time version of the Lagrange chain.
A natural extension of our discretizations could be the construction of a suitable approach for models with a trigonometric or elliptic dependence on the spectral parameter instead of a rational one. To the best of our knowledge there are no results in this direction in literature. We remark here that integrable discretizations for the flows (3.15)-(3.16) have been found by Yu.B. Suris, see [27,28,29], by using rational Lax matrices.