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The Reduced Euler-Lagrange Equations

Marsden, Jerrold E. and Scheurle, Jürgen (1993) The Reduced Euler-Lagrange Equations. In: Dynamics and Control of Mechanical Systems: The Falling Cat and Related Problems. Fields Institute Communications. No.1. AMS and Fields Institute , pp. 139-164. ISBN 9780821892008 http://resolver.caltech.edu/CaltechAUTHORS:20100908-112639904

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Abstract

Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard Euler-Lagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduction of a Lagrangian system on a level set of a conserved quantity, a key object is the Routhian, which is the Lagrangian minus the mechanical connection paired with the fixed value of the momentum map. For unconstrained systems, we use a velocity shifted Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton’s variational principle for the Euler-Lagrange equations breaks up into two sets of equations that represent a set of Euler-Lagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the Euler-Poincar´e equations for the vertical variations. This new set of equations is what we call the reduced Euler-Lagrange equations, and it includes the Euler-Poincaré and the Hamel equations as special cases. We illustrate this methodology for a rigid body with internal rotors and for a particle moving in a magnetic field.


Item Type:Book Section
Additional Information:© 1993, AMS. August, 1992—this version: May 8, 1994. Research partially supported by the Fields Institute, NSF grant DMS-89-22704 and DOE Contract DE-FGO3-92ER25129.
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Funding AgencyGrant Number
Fields InstituteUNSPECIFIED
NSFDMS-89-22704
Department of EnergyDE-FGO3-92ER25129
Record Number:CaltechAUTHORS:20100908-112639904
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100908-112639904
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19829
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:15 Sep 2010 21:35
Last Modified:26 Dec 2012 12:24

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