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Lagrangian Reduction by Stages

Cendra, Hernán and Marsden, Jerrold E. and Ratiu, Tudor S. (2001) Lagrangian Reduction by Stages. Memoirs of the American Mathematical Society, 152 (722). pp. 1-108. ISSN 0065-9266. http://resolver.caltech.edu/CaltechAUTHORS:20100727-102455039

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Abstract

This booklet studies the geometry of the reduction of Lagrangian systems with symmetry in a way that allows the reduction process to be repeated; that is, it develops a context for Lagrangian reduction by stages. The Lagrangian reduction procedure focuses on the geometry of variational structures and how to reduce them to quotient spaces under group actions. This philosophy is well known for the classical cases, such as Routh reduction for systems with cyclic variables (where the symmetry group is Abelian) and Euler{Poincare reduction (for the case in which the conguration space is a Lie group) as well as Euler-Poincare reduction for semidirect products. The context established for this theory is a Lagrangian analogue of the bundle picture on the Hamiltonian side. In this picture, we develop a category that includes, as a special case, the realization of the quotient of a tangent bundle as the Whitney sum of the tangent of the quotient bundle with the associated adjoint bundle. The elements of this new category, called the Lagrange{Poincare category, have enough geometric structure so that the category is stable under the procedure of Lagrangian reduction. Thus, reduction may be repeated, giving the desired context for reduction by stages. Our category may be viewed as a Lagrangian analog of the category of Poisson manifolds in Hamiltonian theory. We also give an intrinsic and geometric way of writing the reduced equations, called the Lagrange{Poincare equations, using covariant derivatives and connections. In addition, the context includes the interpretation of cocycles as curvatures of connections and is general enough to encompass interesting situations involving both semidirect products and central extensions. Examples are given to illustrate the general theory. In classical Routh reduction one usually sets the conserved quantities conjugate to the cyclic variables equal to a constant. In our development, we do not require the imposition of this constraint. For the general theory along these lines, we refer to the complementary work of Marsden, Ratiu and Scheurle [2000], which studies the Lagrange-Routh equations.


Item Type:Article
Additional Information:© 2001 AMS. Received by the editor Received by the editor April 12, 1999 and in revised form March 15, 2000. Updated Oct. 2009. The research of HC was partly done during a sabbatical stay in Control and Dynamical Systems at the California Institute of Technology. The research of JEM was supported in part by the California Institute of Technology and NSF Grants DMS-9802106 and ATM-9873133. The research of TSR was supported in part by NSF Grant DMS-9802378 and FNS Grant 21-54138.98.
Funders:
Funding AgencyGrant Number
California Institute of TechnologyUNSPECIFIED
NSFDMS-9802106
NSFATM-9873133
NSFDMS-9802378
FNS21-54138.98
Subject Keywords:Lagrangian reduction, mechanical systems, variational principles, symmetry.
Classification Code:MSC: Primary 37J15; Secondary 70H33, 53D20.
Record Number:CaltechAUTHORS:20100727-102455039
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100727-102455039
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19199
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:27 Jul 2010 20:31
Last Modified:26 Dec 2012 12:16

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