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

@article{Holm1998TheEE,
  title={The Euler–Poincar{\'e} Equations and Semidirect Products with Applications to Continuum Theories},
  author={Darryl D. Holm and Jerrold E. Marsden and Tudor S. Ratiu},
  journal={Advances in Mathematics},
  year={1998},
  volume={137},
  pages={1-81}
}
We study Euler–Poincare 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–Poincare 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… 
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