Discrete Euler-Poincare and Lie-Poisson equations

In this paper, discrete analogues of Euler-Poincare and Lie-Poisson reducti on theory are developed for systems on finite dimensional Lie groups G with Lagrangians L : TG --> R that are G-invariant. These discrete equations pr ovide 'reduced' numerical algorithms which manifestly preserve the symplect ic structure. The manifold G x G is used as an approximation of TG, and a d iscrete Langragian L : G x G --> R is constructed in such a way that the G- invariance property is preserved. Reduction by G results in a new 'variatio nal' principle for the reduced Lagrangian l : G --> R, and provides the dis crete Euler-Poincare (DEP) equations. Reconstruction of these equations rec overs the discrete Euler-Lagrange equations developed by Marsden et al (Mar sden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys. 199 351-395) an d Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 2 23-246) which are naturally symplectic-momentum algorithms. Furthermore, th e solution of the DEP algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is shown that when G = SO(n), the DEP and DLP algorith ms for a particular choice of the discrete Lagrangian L are equivalent to t he Moser-Veselov scheme for the generalized rigid body.