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Reduction theory and the Lagrange–Routh equations

Marsden, Jerrold E. and Ratiu, Tudor S. and Scheurle, Jürgen (2000) Reduction theory and the Lagrange–Routh equations. Journal of Mathematical Physics, 41 (6). pp. 3379-3429. ISSN 0022-2488. doi:10.1063/1.533317.

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Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré, and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles, along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, to stability theory, integrable systems, as well as geometric phases. This paper surveys progress in selected topics in reduction theory, especially those of the last few decades as well as presenting new results on non-Abelian Routh reduction. We develop the geometry of the associated Lagrange–Routh equations in some detail. The paper puts the new results in the general context of reduction theory and discusses some future directions.

Item Type:Article
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Ratiu, Tudor S.0000-0003-1972-5768
Additional Information:© 2000 American Institute of Physics. Received 8 November 1999; accepted 10 March 2000. We thank our many collaborators and students for their help, direct or indirect, with this paper. In particular, we would like to single out Anthony Blaom, Anthony Bloch, Sameer Jalnapurkar, Hans-Peter Kruse, Melvin Leok, Naomi Leonard, Sergey Pekarsky, Matt Perlmutter, Steve Shkoller, Alan Weinstein, and Matt West. Research by J.E.M. was partially supported by the National Science Foundation, the Humboldt Foundation, and the California Institute of Technology. Research by T.S.R. was partially supported by the US and Swiss National Science Foundations and the Humboldt Foundation.
Funding AgencyGrant Number
Humboldt FoundationUNSPECIFIED
Swiss National Science FoundationUNSPECIFIED
Subject Keywords:classical mechanics, symmetry, conservation laws, variational techniques, geometry
Issue or Number:6
Classification Code:PACS: 45.05.+x; 45.10.Db; 45.10.Na; 02.30.Xx; 02.30.Yy; 02.40.-k
Record Number:CaltechAUTHORS:20101005-152140019
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Official Citation:Reduction theory and the Lagrange--Routh equations Jerrold E. Marsden, Tudor S. Ratiu, and Jurgen Scheurle, J. Math. Phys. 41, 3379 (2000), DOI:10.1063/1.533317
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20312
Deposited By: Tony Diaz
Deposited On:16 Nov 2010 18:55
Last Modified:08 Nov 2021 23:58

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