Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 398

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

A. I. Bobenko and Y. B. Suris

Letters in Math. Phys., 1999, to appear

A discrete version of Lagrangian reduction is developed in the context ofdiscrete time Lagrangian systems on $G imes G$, where $G$ is a Lie group.We consider the case when the Lagrange function is invariant with respect to theaction of an isotropy subgroup of a fixed element in the representationspace of $G$. In this context the reduction of the discrete Euler--Lagrangeequations 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 equationsleads to discrete Hamiltonian (Lie--Poisson) systems on a dual space to a semiproduct Lie algebra.

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