Discrete Lagrangian Reduction, Discrete EulerPoincar Equations, and Semidirect Products

Adiscrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G G, where G is a Lie group. We consider the case when the Lagrange function is invariantwith respect to the action ofan isotropysubgroup ofa¢xed 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^Poincarë equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler^Poincarë equations leads to discrete Hamiltonian (Lie^Poisson) systems on a dual space to a semiproduct Lie algebra. Mathematics Subject Classi¢cations (1991): 58F05, 39A12, 70H15. Key words: Lagrangian systems on Lie groups, di¡erence equations, Lagrangian reduction, discretization. 1. Introduction Dynamical systems with symmetry play an important role in the mathematical modelling of a vast variety of physical and mechanical processes. A Hamiltonian approach to such systems is nowadays a well-established theory [1, 2, 4, 8, 9, 13]. More recently, also a variational (Lagrangian) description of systems with symmetries has also attracted much attention [5, 6, 10]. In particular, in [6] and [10] the corresponding theory was developed for Lagrangian systems on Lie groups, i.e. for Lagrangians de¢ned on tangent bundles TG of Lie groups. A symmetry of the Lagrangian with respect to a subgroup action leads to a reduced system on a semidirect product described by the so-called Euler^Poincarë equation. In this Letter we develop a discrete analog of this theory, i.e. for Lagrangians de¢ned on G G. We introduce the corresponding reduced systems and derive the discrete Euler^Poincarë equations. We establish symplectic properties of the corresponding discrete dynamical systems. The continuous time theory may be considered as a limiting case of the discrete time one. An important particular case, when the representation of G participating in the general theory is chosen to be the adjoint representation, is developed in [3]. Letters in Mathematical Physics 49: 79^93, 1999. 79 # 1999 Kluwer Academic Publishers. Printed in the Netherlands. 2. Lagrangian Mechanics on TG and on G G Recall that a continuos time Lagrangian system is de¢ned by a smooth function L…g; _ g† : TG 7! R on the tangent bundle of a smooth manifold G. The function L is called the Lagrange function. We will be dealing here only with the case when G carries an additional structure of a Lie grioup. For an arbitrary function g…t† : ‰t0; t1Š 7! G one can consider the action functional