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Published 1998 | Published
Journal Article Open

Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products

Abstract

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible uids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one nds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given.

Additional Information

© 1998 American Mathematical Society. Received February 1997; this version, October 8, 1997. Research partially supported by NSF grant DMS 96-33161 and DOE contract DE-FG0395-ER25251. Research partially supported by NSF Grant DMS-9503273 and DOE contract DE-FG03- 95ER25245-A000.

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August 19, 2023
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