Action principle and the Hamiltonian formulation for the Maxwell–Vlasov equations on a symplectic leaf

Abstract

An action principle for the Maxwell–Vlasov (MV) equation is formulated in terms of the Maxwell fields and the generating function, w(z,t) for deviations from a reference distribution function, f0(z), which labels a symplectic leaf. New formal fields suitable for variations are defined. These fields give rise to a symplectic and Poisson structure. The Hamiltonian formulation of the equations is found in terms of the new formal fields, and it is found how to derive Larsson’s action principle [J. Plasma Phys. 48, 13 (1992); ibid. 49, 255 (1993)] and generalized versions of it on a Lagrangian constraint manifold in a double symplectic space. It is also shown how the relativistic Maxwell–Vlasov system and the Maxwell–Vlasov system with a time-dependent reference state can be formulated as an action principle and Hamiltonian system in terms of eight-dimensional particle phase space coordinates.