On the Hamilton-Jacobi variational formulation of the Vlasov equation

The Vlasov-Poisson and Vlasov-Maxwell equations possess various variational formulations or action principles. I will discuss a particular variational principle that is based on a Hamiltonian-Jacobi formulation of Vlasov theory, a formulation that is not widely known. I will show how this formulation can be reduced for describing the Vlasov-Poisson system. The resulting system is of Hamilton-Jacobi form, but with nonlinear global coupling to the Poisson equation. A description of phase (function)space geometry and relation to Hamilton-Jacobi PDE methods and weak KAM will be given.

H. Ye and P. J. Morrison Phys. Fluids 4B, 771 (1992). D. Pfirsch, Z. Naturforsch. 39a, 1 (1984); D. Pfirsch and P. J. Morrison, Phys. Rev. 32A, 1714 (1985).