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

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G x 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. Within this context, the reduction of the discrete Euler-Lagrange equations is shown to lead to the so-called discrete Euler-Poincare equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler-Poincare equations leads to discrete Hamiltonian (Lie-Poisson) systems on a dual space to a semiproduct Lie algebra.