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

Alexander I. Bobenko·Yuri B. Suris
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Abstract

A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, 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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Alexander I. Bobenko, Yuri B. Suris. 1999-06-15. Discrete Lagrangian reduction, discrete Euler-Poincare equations, and semidirect products. https://arxiv.org/abs/math/9906108

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