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Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators

Zhong, Ge and Marsden, Jerrold E. (1988) Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Physics Letters A, 133 (3). pp. 134-139. ISSN 0375-9601. doi:10.1016/0375-9601(88)90773-6.

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We present results on numerical integrators that exactly preserve momentum maps and Poisson brackets, thereby inducing integrators that preserve the natural Lie-Poisson structure on the duals of Lie algebras. The techniques are baseda on time-stepping with the generating function obtained as an approximate solution to the Hamilton-Jacobi equation, following ideas of deVogelaére, Channel,, and Feng. To accomplish this, the Hamilton-Jacobi theory is reduced from T*G to g*, where g is the Lie algebra of a Lie group G. The algorithms exactly preserve any additional conserved quantities in the problem. An explicit algorithm is given for any semi-simple group and in particular for the Euler equation of rigid body dynamics.

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Additional Information:© 1988 Published by Elsevier. Received 5 May 1988; accepted 27 June 1988 Communicated by D.D. Holm Available online 18 September 2002. This research was partially supported by DOE contract DEAT03-85ER 12097 and the Mathematical Sciences Institute at Cornell. It is a pleasure to thank Rene deVogelaere, Bob Grossman, Darryl Holm, Feng Kang, Swan Kim, P,S, Krishnaprasad, Debbie Lewis, Clint Scovel, Juan Simo, and Alan Weinstein for useful discussions. We would especially like to thank Paul Channell for a careful reading of the manuscript and for informing us of the implementation of algorithms of this sort.
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Department of EnergyDEAT03-85ER 12097
Mathematical Sciences Institute at CornellUNSPECIFIED
Issue or Number:3
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ID Code:19354
Deposited By: Ruth Sustaita
Deposited On:10 Aug 2010 20:42
Last Modified:08 Nov 2021 23:51

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