Discrete Lagrangian reduction, discrete Euler-Poincare equations, and semidirect products

1999/06/15 by Alexander I. Bobenko, Yuri B. Suris
Mathematics · Physics and Astronomy · #math.SG #math-ph #math.DS #math.MP

paper · pdf

published as Lett. Math. Phys., 1999, V. 49, p.79-93. · 16 pp., LaTeX

arxiv created 1999/06/15 · arxiv updated 2009/11/30

Abstract

A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on G× G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. In this context the reduction of the discrete Euler-Lagrange equations is shown to lead to the so called discrete Euler-Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler-Poincaré equations leads to discrete Hamiltonian (Lie-Poisson) systems on a dual space to a semiproduct Lie algebra.

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