Symmetry reduction of discrete Lagrangian mechanics on Lie groups

Abstract

For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie–Poisson structure on the dual of the Lie algebra g^* by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ωL on G×G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton–Jacobi equation are made. The rigid body is discussed as an example.