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Hamilton–Pontryagin integrators on Lie groups part I: introduction and structure-preserving properties

Bou-Rabee, Nawaf and Marsden, Jerrold E. (2009) Hamilton–Pontryagin integrators on Lie groups part I: introduction and structure-preserving properties. Foundations of Computational Mathematics, 9 (2). pp. 197-219. ISSN 1615-3375.

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In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge– Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Störmer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structurepreserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.

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Additional Information:© SFoCM 2008. Received: 5 February 2007. Revised: 7 January 2008. Accepted: 29 February 2008. Published online: 15 May 2008. Research partially supported by the National Science Foundation through NSF grant DMS-0204474. Mathematics Subject Classification (2000) 37M15 · 58E30 · 65P10 · 70EXX · 70HXX
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Subject Keywords:Variational integrators; Hamilton Pontryagin; Lie group integrators
Record Number:CaltechAUTHORS:20090811-135447283
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14968
Deposited By: Ruth Sustaita
Deposited On:11 Aug 2009 23:10
Last Modified:26 Dec 2012 11:10

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