Variational principles for spin systems and the Kirchhoff rod

François Gay-Balma; Darryl D. Holm; Tudor S. Ratiu

doi:10.3934/jgm.2009.1.417

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2009, Volume 1, Issue 4: 417-444. Doi: 10.3934/jgm.2009.1.417

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Variational principles for spin systems and the Kirchhoff rod

1.
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 911125
2.
Department of Mathematics and Institute for Mathematical Sciences, Imperial College, London, SW7 2AZ
3.
Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015

Received: March 31, 2009

Revised: June 30, 2009

Published: November 2009

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Abstract

We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.

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Mathematics Subject Classification: Primary: 70H33, 70G65, 70G75; Secondary: 74K10, 82D30.

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