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