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Mechanical Integrators Derived from a Discrete Variational Principle

Wendlandt, Jeffrey M. and Marsden, Jerrold E. (1997) Mechanical Integrators Derived from a Discrete Variational Principle. Physica D, 106 (3-4). pp. 223-246. ISSN 0167-2789. doi:10.1016/S0167-2789(97)00051-1.

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Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this paper, we present a procedure to construct time-stepping algorithms that approximate the flow of continuous ODEs for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. The discrete equations share similarities to the continuous equations by preserving invariants, including the symplectic form and the momentum map. We first present a formulation of discrete mechanics along with a discrete variational principle. We then show that the resulting equations of motion preserve the symplectic form and that this formulation of mechanics leads to conservation laws from a discrete version of Noether's theorem. We then use the discrete mechanics formulation to develop a procedure for constructing symplectic-momentum mechanical integrators for Lagrangian systems with holonomic constraints. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.

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Additional Information:© 1997 Published by Elsevier Science B.V. Preprint submitted to Elsevier Preprint, 23 February 1997. Received 17 September 1996; revised 7 March 1997; accepted 18 March 1997; Communicated by J.D. Meiss Available online 15 May 1998. Research partially supported by DOE contract DE-FG03095ER-25251 Research partially supported by DOE contract DE-FG03095ER-25251 and the California Institute of Technology. We first would like to thank Andrew Lewis for pointing out [3]. We would also like to thank Richard Murray and Abhi Jain for help with an initial investigation into mechanical integrators. We appreciate the useful comments and discussions provided by Francisco Armero and Oscar Gonzalez. We also thank Robert MacKay and Shmuel Weissman for useful discussions.
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG0395-ER25251
Subject Keywords:Discrete mechanics; Symplectic integration; Symplectic-momentum integration; Rigid body integration
Issue or Number:3-4
Classification Code:PACS: 46.10, 02.60
Record Number:CaltechAUTHORS:20100917-091859665
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20003
Deposited By: Ruth Sustaita
Deposited On:17 Sep 2010 18:21
Last Modified:08 Nov 2021 23:56

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