Marsden, Jerrold E. and Scheurle, Jürgen (1993) The Reduced EulerLagrange Equations. In: Dynamics and Control of Mechanical Systems: The Falling Cat and Related Problems. Fields Institute Communications. No.1. AMS and Fields Institute , pp. 139164. ISBN 9780821892008 http://resolver.caltech.edu/CaltechAUTHORS:20100908112639904

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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 EulerLagrange 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 EulerLagrange equations breaks up into two sets of equations that represent a set of EulerLagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the EulerPoincar´e equations for the vertical variations. This new set of equations is what we call the reduced EulerLagrange equations, and it includes the EulerPoincaré 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 DMS8922704 and DOE Contract DEFG0392ER25129.  
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Record Number:  CaltechAUTHORS:20100908112639904  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20100908112639904  
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Deposited By:  Ruth Sustaita  
Deposited On:  15 Sep 2010 21:35  
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