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Discrete Routh reduction

Jalnapurkar, Sameer M. and Leok, Melvin and Marsden, Jerrold E. and West, Matthew (2006) Discrete Routh reduction. Journal of Physics A: Mathematical and General, 39 (19). pp. 5521-5544. ISSN 0305-4470. http://resolver.caltech.edu/CaltechAUTHORS:JALjpa06

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Abstract

This paper develops the theory of Abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with Abelian symmetry. The reduction of variational Runge–Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical J2 correction, as well as the double spherical pendulum. The J2 problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a non-trivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the non-canonical nature of the symplectic structure.


Item Type:Article
Additional Information:Copyright © Institute of Physics and IOP Publishing Limited 2006. Received 24 August 2005, in final form 5 January 2006, Published 24 April 2006, Print publication: Issue 19 (12 May 2006) We gratefully acknowledge helpful comments and suggestions of Alan Weinstein and the referees. SMJ was supported in part by ISRO and DRDO through the Nonlinear Studies Group, Indian Institute of Science, Bangalore. ML was supported in part by NSF Grant DMS-0504747, and a faculty grant and fellowship from the Rackham Graduate School, University of Michigan. JEM was supported in part by NSF ITR Grant ACI-0204932. SPECIAL ISSUE ON GEOMETRICAL NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS, Journal of Mathematical Physics A Volume 39, Number 19, 12 May 2006 http://www.iop.org/EJ/toc/0305-4470/39/19
Record Number:CaltechAUTHORS:JALjpa06
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:JALjpa06
Alternative URL:http://dx.doi.org/10.1088/0305-4470/39/19/S12
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:3990
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:21 Jul 2006
Last Modified:26 Dec 2012 08:57

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