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Reduction, Symmetry and Phases in Mechanics

Marsden, J. E. and Montgomery, R. and Ratiu, T. S. (1990) Reduction, Symmetry and Phases in Mechanics. Memoirs of the American Mathematical Society, 88 (436). ISSN 0065-9266.

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Various holonomy phenomena are shown to be instances of the reconstruction procedure for mechanical systems with symmetry. We systematically exploit this point of view for fixed systems (for example with controls on the internal, or reduced, variables) and for slowly moving systems in an adiabatic context. For the latter, we obtain the phases as the holonomy for a connection which synthesizes the Cartan connection for moving mechanical systems with the Hannay-Berry connection for integrable systems. This synthesis allows one to treat in a natural way examples like the ball in the slowly rotating hoop and also non-integrable mechanical systems.

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Ratiu, T. S.0000-0003-1972-5768
Additional Information:© 1990 American Mathematical Society. Manuscript Received April 10, 1989 In Revised Form: January 5,1990. We thank J. Anandan, A. Fischer, J. Koiller, P. Krishnaprasad, R. Littlejohn, A. Pines, A. Weinstein, and the referee for their valuable comments. J.E. Marsden, Department of Mathematics, University of California, Berkeley, CA 94720. Research partially supported by NSF grant DMS 8702502 and DOE Contract DE-AT03-88ER-12097. R. Montgomery, Department of Mathematics, University of California, Santa Cruz, CA 95064. Research partially supported by an NSF postdoctoral fellowship and NSF grant DMS 8702502. T. Ratiu, Department of Mathematics, University of California, Santa Cruz, CA 95064. Research partially supported by NSF grant DMS 8701318-01, and AFOSRJDARPA contract F49620-87-C- 0118. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F, 70H.
Funding AgencyGrant Number
NSFDMS 8702502
Department of Energy (DOE)DE-AT03-88ER-12097
NSFDMS 8702502
NSFDMS 8701318-01
Air Force Office of Scientific Research (AFOSR)F49620-87-C- 0118
NSF Postdoctoral FellowshipUNSPECIFIED
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Series Name:Memoirs of the American Mathematical Society
Issue or Number:436
Classification Code:MSC: Primary 58F, 70H.
Record Number:CaltechAUTHORS:20100902-092616867
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19765
Deposited By: Ruth Sustaita
Deposited On:04 Sep 2010 03:21
Last Modified:09 Mar 2020 13:19

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