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December  2009, 1(4): 417-444. doi: 10.3934/jgm.2009.1.417

Variational principles for spin systems and the Kirchhoff rod

François Gay-Balma 1, , Darryl D. Holm 2, and  Tudor S. Ratiu 3,

1. 

Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 911125, United States
2. 

Department of Mathematics and Institute for Mathematical Sciences, Imperial College, London, SW7 2AZ, United Kingdom
3. 

Section de Mathématiques and Bernoulli Center, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland

Received  April 2009 Revised  July 2009 Published  January 2010

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.
Keywords: Kirchhoff rod., affine Euler-Poincaré equations, Lagrangian reducton, spin system, variational principles, cocycle, symmetry.
Mathematics Subject Classification: Primary: 70H33, 70G65, 70G75; Secondary: 74K10, 82D3.
Citation: François Gay-Balma, Darryl D. Holm, Tudor S. Ratiu. Variational principles for spin systems and the Kirchhoff rod. Journal of Geometric Mechanics, 2009, 1 (4) : 417-444. doi: 10.3934/jgm.2009.1.417
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