THE EULER-POINCARE EQUATIONS AND SEMIDIRECT PRODUCTS WITH APPLICATIONS TO CONTINUUM-THEORIES

We study Euler-Poincare systems (i.e., the Lagrangian analogue of Lie- Poisson Hamiltonian systems) defined on semidirect product Lie algebra s. We first give a derivation of the Euler-Poincare equations for a pa rameter dependent Lagrangian by using a variational principle of Lagra nge d'Alembert type. Then we derive an abstract Kelvin-Noether theorem for these equations. We also explore their relation with the theory o f Lie-Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is ofte n not invertible; thus, it does not produce a corresponding Euler-Poin care system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler-Poincare systems entirely within the L agrangian 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 e quations, which have many potentially interesting analytical propertie s. These equations are Euler-Poincare equations for geodesics on diffe omorphism groups (in the sense of the Arnold program) but where the me tric is H-1 rather than L-2. (C) 1998 Academic Press.