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The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

Holm, Darryl D. and Marsden, Jerrold E. and Ratiu, Tudor S. (1998) The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories. Advances in Mathematics, 137 (1). pp. 1-81. ISSN 0001-8708.

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We study Euler–Poincaré systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract Kelvin–Noether theorem for these equations. We also explore their relation with the theory of Lie–Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa–Holm equations, which have many potentially interesting analytical properties. These equations are Euler–Poincaré equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather thanL^2.

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Additional Information:© 1998 Academic Press. All rights reserved. Received 17 November 1997; accepted 3 January 1998. ; Available online 15 April 2002. Research partially supported by NSF grant DMS 96–33161. Research partially supported by NSF Grant DMS-9503273 and DOE contract DE-FG03-95ER25245-A000. We thank Hernan Cendra, Shiyi Chen, Ciprian Foias, Mark Hoyle, David Levermore, Len Margolin, Gerard Misiolek, Balu Nadiga, Matthew Perlmutter, Steve Shkoller and Edriss Titi for valuable discussions and remarks.
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NSFDMS 96–33161
Department of Energy (DOE)DE-FG03-95ER25245-A000
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19410
Deposited By: Ruth Sustaita
Deposited On:13 Aug 2010 16:53
Last Modified:08 May 2015 17:43

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