Discrete Euler-Poincar\'{e} and Lie-Poisson Equations

In this paper, discrete analogues of Euler-Poincar\'{e} and Lie-Poisson reduction theory are developed for systems on finite dimensional Lie groups $G$ with Lagrangians $L:TG \to {\mathbb R}$ that are $G$-invariant. These discrete equations provide ``reduced'' numerical algorithms which manifestly preserve the symplectic structure. The manifold $G \times G$ is used as an approximation of $TG$, and a discrete Langragian ${\mathbb L}:G \times G \to {\mathbb R}$ is construced in such a way that the $G$-invariance property is preserved. Reduction by $G$ results in new ``variational'' principle for the reduced Lagrangian $\ell:G \to {\mathbb R}$, and provides the discrete Euler-Poincar\'{e} (DEP) equations. Reconstruction of these equations recovers the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when $G=\text{SO} (n)$, the DEP and DLP algorithms for a particular choice of the discrete Lagrangian ${\mathbb L}$ are equivalent to the Moser-Veselov scheme for the generalized rigid body. %As an application, a reduced symplectic integrator for two dimensional %hydrodynamics is constructed using the SU$(n)$ approximation to the volume %preserving diffeomorphism group of ${\mathbb T}^2$.