Abstract
Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg chain with classical spins and a new discrete system on the Stiefel manifold. The integrability is shown with the help of a Lax-pair representation which is found via a factorization of certain matrix polynomials. The complete description of the dynamics is given in terms of Abelian functions; the flow becomes linear on a Prym variety corresponding to a spectral curve. The approach is also applied to the billiard problem in the interior of anN-dimensional ellipsoid.
Highlights
In this paper we study a class of discrete integrable systems which are closely related to problems occurring in mathematical physics such as the Heisenberg model for classical spins or the billiard problem in the interior of an ellipsoid
A discrete system can be viewed as the iterates of a symplectic mapping, the time t EΈ being the number of iterations
Such a system will be called integrable if it possesses sufficiently many integrals which are in involution with respect to a symplectic structure. To describe such a discrete system we take as starting point a variational principle for a functional S = S(X) defined on the space of sequences X = (Xk), keΈ by a formal sum s= Σ &{xk,xk+1). keZ
Summary
In this paper we study a class of discrete integrable systems which are closely related to problems occurring in mathematical physics such as the Heisenberg model for classical spins or the billiard problem in the interior of an ellipsoid. In the problem of orthogonal chains we are able to describe such a class of matrix polynomials and a corresponding factorization which can be made unique by specifying a suitable splitting of the spectrum From this Lax representation we will find the integrals as well as the algebraic curve on whose Jacobian variety the flow becomes linear in k. We have to solve two crucial problems: a) to define the mapping φ in a unique way by selecting a branch of the correspondence and b) to verify that this mapping is integrable Both these problems can be reduced to an appropriate factorization problem for a matrix polynomial, as we will show now.
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