The Euler-Poincaré Equations and Semidirect Products
with Applications to Continuum Theories
Holm, D. D., J. E. Marsden and T. S. Ratiu
Adv. in Math., 137, 1-81
Abstract:
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 H1 rather than L2.
