  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 Lie algebra. The Legendre
transformation in such cases is often not invertible; so it does not produce a
corresponding Euler-Poincare system on that Lie algebra. We avoid this
potential difficulty by developing the theory of Euler-Poincare 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-Poincare equations for geodesics on diffeomorphism groups
(in the sense of the Arnold program) but where the metric is H^1 rather than
L^2.
