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Discrete versions of some classical integrable systems and factorization of matrix polynomials

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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.

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Authors and Affiliations

  1. Forschungsinstitut für Mathematik, ETH Zürich, CH-8092, Zürich, Switzerland

    Jürgen Moser

  2. Moscow State University, Moscow, USSR

    Alexander P. Veselov

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  1. Jürgen Moser
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  2. Alexander P. Veselov
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Communicated by J. N. Mather

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Moser, J., Veselov, A.P. Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun.Math. Phys. 139, 217–243 (1991). https://doi.org/10.1007/BF02352494

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  • DOI: https://doi.org/10.1007/BF02352494

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