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

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 isH1rather thanL2. © 1998 Academic Press.