Discrete Euler-Poincaré and Lie-Poisson equations

Abstract

In this paper, discrete analogues of Euler–Poincare and Lie–Poisson reduction theory are developed for systems on finite dimensional Lie groups G with Lagrangians L : TG → R that are G-invariant. These discrete equations provide ‘reduced’ numerical algorithms which manifestly preserve the symplectic structure. The manifold G x G is used as an approximation of TG, and a discrete 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 ‘variational’ principle for the reduced Lagrangian l : G → R, and provides the discrete Euler–Poincare (DEP) equations. Reconstruction of these equations recovers the discrete Euler–Lagrange equations developed by Marsden et al (Marsden J E, Patrick G and Shkoller S 1998 Commun. Math. Phys. 199 351–395) and Wendlandt and Marsden (Wendlandt J M and Marsden J E 1997 Physica D 106 223–246) 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 = SO(n), the DEP and DLP algorithms for a particular choice of the discrete Lagrangian L are equivalent to the Moser–Veselov scheme for the generalized rigid body.