99-277 A.I. Bobenko, Yu.B. Suris

Discrete Lagrangian reduction, discrete Euler--Poincar\'e equations, and semidirect products (461K, Postscript) Jul 23, 99

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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 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.

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