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The Jacobi-Maupertuis Principle in Variational Integrators

Nair, Sujit and Ober-Blöbaum, Sina and Marsden, Jerrold E. (2009) The Jacobi-Maupertuis Principle in Variational Integrators. In: Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics 2009. AIP Conference Proceedings. No.1168. American Institute of Physics , Melville, NY, pp. 464-467. ISBN 978-0-7354-0709.

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In this paper, we develop a hybrid variational integrator based on the Jacobi-Maupertuis Principle of Least Action. The Jacobi-Maupertuis principle states that for a mechanical system with total energy E and potential energy V(q), the curve traced out by the system on a constant energy surface minimizes the action given by ∫√[2(E-V(q))] ds where ds is the line element on the constant energy surface with respect to the kinetic energy of the system. The key feature is that the principle is a parametrization independent geodesic problem. We show that this principle can be combined with traditional variational integrators and can be used to efficiently handle high velocity regions where small time steps would otherwise be required. This is done by switching between the Hamilton principle and the Jacobi-Maupertuis principle depending upon the kinetic energy of the system. We demonstrate our technique for the Kepler problem and discuss some ongoing and future work in studying the energy and momentum behavior of the resulting integrator.

Item Type:Book Section
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Additional Information:© 2009 American Institute of Physics. Issue Date: 9 September 2009.
Subject Keywords:Jacobian matrices; partial differential equations; discrete Fourier transforms; variational integrator; discrete mechanics; Jacobi-Maupertius principle
Series Name:AIP Conference Proceedings
Issue or Number:1168
Classification Code:PACS: 02.40.Yy, 02.60.Cb, 45.10.Db
Record Number:CaltechAUTHORS:20100125-115309328
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Official Citation:The Jacobi-Maupertuis Principle in Variational Integrators Sujit Nair, Sina Ober-Blobaum, and Jerrold E. Marsden, AIP Conf. Proc. 1168, 464 (2009), DOI:10.1063/1.3241498
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:17303
Deposited By: Jason Perez
Deposited On:28 Jan 2010 19:19
Last Modified:03 Oct 2019 01:25

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