Discrete Routh reduction

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dc.contributor.author Jalnapurkar, Sameer M. en_US
dc.contributor.author Leok, Melvin en_US
dc.contributor.author Marsden, Jerrold E. en_US
dc.contributor.author West, Matthew en_US
dc.date.accessioned 2006-12-19T18:49:00Z
dc.date.available 2006-12-19T18:49:00Z
dc.date.issued 2006-05-12 en_US
dc.identifier.citation Jalnapurkar, Sameer M; Leok, Melvin; Marsden, Jerrold E; West, Matthew (2006). "Discrete Routh reduction." Journal of Physics A: Mathematical and General. 39(19): 5521-5544. <http://hdl.handle.net/2027.42/48794> en_US
dc.identifier.issn 0305-4470 en_US
dc.identifier.uri http://hdl.handle.net/2027.42/48794
dc.description.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. en_US
dc.format.extent 3118 bytes
dc.format.extent 647130 bytes
dc.format.mimetype text/plain
dc.format.mimetype application/pdf
dc.language.iso en_US
dc.publisher IOP Publishing Ltd en_US
dc.title Discrete Routh reduction en_US
dc.type Article en_US
dc.subject.hlbsecondlevel Physics en_US
dc.subject.hlbtoplevel Science en_US
dc.description.peerreviewed Peer Reviewed en_US
dc.contributor.affiliationum Department of Mathematics, University of Michigan, East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA en_US
dc.contributor.affiliationother Department of Mathematics, Indian Institute of Science, Bangalore, India en_US
dc.contributor.affiliationother Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125-8100, USA en_US
dc.contributor.affiliationother Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305-4035, USA en_US
dc.contributor.affiliationumcampus Ann Arbor en_US
dc.description.bitstreamurl http://deepblue.lib.umich.edu/bitstream/2027.42/48794/2/a6_19_s12.pdf en_US
dc.identifier.doi http://dx.doi.org/10.1088/0305-4470/39/19/S12 en_US
dc.identifier.source Journal of Physics A: Mathematical and General. en_US
dc.owningcollname Interdisciplinary and Peer-Reviewed
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