On the Hamilton-Jacobi Variational Formulation of the Vlasov Equation

In this note we briefly review the Hamilton-Jacobi (HJ) formulation of Vlasov-like systems. This is a general formulation that applies to the Maxwell-Vlasov system and various guiding center and gyrokinetic theories with any number of species. It applies to both nonrelativistic and relativistic versions of these theories and even to the Vlasov-Einstein system. Indeed, it is quite general and applies to any Vlasov-like theory, but we will review it in its simplest context of the Vlasov-Poisson system. The formulation evolved out of early work of Pfirsch [1], but the general formulation was first given in [2]. The HJ formulation is variational – it has in fact two action principles, and so it provides a natural method via Noether’s theorem for obtaining unambiguous energymomentum tensors for general kinetic theories. These were obtained and discussed in a sequence of papers [2–4] and this work was continued in [5, 6], where errors in the literature were pointed out. This note is organized as follows. In Sec. II we review the HJ theory in the context of classical mechanics. Then in Sec. III the action principle of [2] for the general theory is described along with a reduced version given in [7]. Finally, in Sec. IV we conclude.