Dd. Holm et al., THE EULER-POINCARE EQUATIONS AND SEMIDIRECT PRODUCTS WITH APPLICATIONS TO CONTINUUM-THEORIES, Advances in mathematics (New York. 1965), 137(1), 1998, pp. 1-81
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.