THE MAXWELL-VLASOV EQUATIONS IN EULER-POINCARE FORM

Low's well-known action principle for the Maxwell-Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture o f Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangia n, as one does for the Euler equations for the rigid body and ideal fl uids, we first transform this action principle into purely Eulerian va riables. Hamilton's principle for the Eulerian description of Low's ac tion principle then casts the Maxwell-Vlasov equations into Euler-Poin care form for right invariant motion on the diffeomorphism group of po sition-velocity phase space, R-6. Legendre transforming the Eulerian f orm of Low's action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Eule r-Poincare equations, this Hamiltonian formulation can be written in t erms of a Poisson structure that contains the Lie-Poisson bracket on t he dual of a semidirect product Lie algebra. Because of degeneracies i n the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell-Vlasov Poisson structure is kn own, whose ingredients are the Lie-Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born-Infeld b rackets for the Maxwell field. We discuss the relationship between the se two Hamiltonian formulations. We also discuss the general Kelvin-No ether theorem for Euler-Poincare equations and its meaning in the plas ma context. (C) 1998 American Institute of Physics.