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Published November 1, 1999 | public
Journal Article Open

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.

Additional Information

© Institute of Physics and IOP Publishing Limited 1999. Received 15 January 1999. Print publication: Issue 6 (November 1999). Recommended by J Laskar. The authors would like to thank Anthony Bloch, Peter Crouch and Tudor Ratiu for helpful comments. SS and SP would like to thank the Center for Nonlinear Science for providing a valuable setting where much of thisworkwas performed. SS and JEM were partially supported by the NSF-KDI grant ATM-98-73133.

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August 22, 2023
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