Integrable Lagrangian correspondences and the factorization of matrix polynomials (Q808056)

This paper studies integrable Lagrangian systems with discrete time. The isospectral transformation method is used. In this method the factorization of matrix polynomials is essential. The method is demonstrated with some examples of integrable discrete systems. They are the stationary problems for classical Heisenberg's chain and its generalization to the case where the variable takes values in appropriate homogeneous spaces. For the group O(N), the discrete analog of Euler- Arnold's problem of motion of a multidimensional rigid body is obtained. Another class of problems are billiards in regions of spaces of constant curvature, restricted by corresponding conic sections.