A.I. Bobenko, Yu.B. Suris
Discrete Lagrangian reduction, discrete Euler--Poincar\'e equations, and semidirect products
(461K, Postscript)

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 theaction 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--Poincar\'e 
equations. A constrained variational principle is derived. 
The Legendre transformation of the discrete Euler--Poincar\'e equations 
leads to discrete Hamiltonian (Lie--Poisson) systems on a dual space 
to a semiproduct Lie algebra.