Low's well known action principle for the Maxwell-Vlasov
equations of ideal plasma dynamics was originally expressed in terms
of a mixture of Eulerian and Lagrangian variables. By imposing
suitable constraints on the variations and analyzing invariance
properties of the Lagrangian, as one does for the Euler equations for
the rigid body and ideal fluids, we first transform this action
principle into purely Eulerian variables. Hamilton's principle for
the Eulerian description of Low's action principle then casts the
Maxwell-Vlasov equations into Euler-Poincaré form for right
invariant motion on the diffeomorphism group of position-velocity
phase space,
6. Legendre transforming the Eulerian form
of Low's action principle produces the Hamiltonian formulation of these
equations in the Eulerian description. Since it arises from
Euler-Poincaré equations, this Hamiltonian formulation can be
written in terms of a Poisson structure that contains the Lie-Poisson
bracket on the dual of a semidirect product Lie algebra. Because of
degeneracies in the Lagrangian, the Legendre transform is dealt with
using the Dirac theory of constraints. Another Maxwell-Vlasov Poisson
structure is known, whose ingredients are the Lie-Poisson bracket on
the dual of the Lie algebra of symplectomorphisms of phase space and
the Born-Infeld brackets for the Maxwell field. We discuss the
relationship between these two Hamiltonian formulations. We also
discuss the general Kelvin-Noether theorem for Euler-Poincaré
equations and its meaning in the plasma context.
