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Mechanical integrators derived from a discrete variational principle

Wendland, Jeffrey M. and Marsden, Jerrold E (1996) Mechanical integrators derived from a discrete variational principle. California Institute of Technology , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechCDSTR:1996.013

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Abstract

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 ODE'S 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 mechanical integrators for continuous Lagrangian systems. We apply the construction procedure to the rigid body and the double spherical pendulum to demonstrate numerical properties of the integrators.


Item Type:Report or Paper (Technical Report)
Additional Information:Research partially supported by DOE contract DE-FG03-95ER-25251 and the California Institute of Technology. We first would like to thank Andrew Lewis for pointing out (Baez and Gilliam, 1995). 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.
Group:Control and Dynamical Systems Technical Reports
Record Number:CaltechCDSTR:1996.013
Persistent URL:http://resolver.caltech.edu/CaltechCDSTR:1996.013
Usage Policy:You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format.
ID Code:28124
Collection:CaltechCDSTR
Deposited By: Imported from CaltechCDSTR
Deposited On:14 Dec 2007
Last Modified:26 Dec 2012 14:31

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