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Discrete Euler-Poincaré and Lie-Poisson equations

Marsden, Jerrold E. and Pekarsky, Sergey and Shkoller, Steve (1999) Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity, 12 (6). pp. 1647-1662. ISSN 0951-7715. doi:10.1088/0951-7715/12/6/314.

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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.

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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.
Issue or Number:6
Record Number:CaltechAUTHORS:MARnonlin99
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:848
Deposited By: Tony Diaz
Deposited On:21 Oct 2005
Last Modified:08 Nov 2021 19:05

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