Action principles for the Vlasov equation

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Abstract
Five action principles for the Vlasov–Poisson and Vlasov–Maxwell equations, which differ by the variables incorporated to describe the distribution of particles in phase space, are presented. Three action principles previously known for the Vlasov–Maxwell equations are altered so as to produce the Vlasov–Poisson equation upon variation with respect to only the particle variables, and one action principle previously known for the Vlasov–Poisson equation is altered to produce the Vlasov–Maxwell equations upon variations with respect to particle and field variables independently. Also, a new action principle for both systems, which is called the leaf action, is presented. This new action has the desirable features of using only a single generating function as the dynamical variable for describing the particle distribution, and manifestly preserving invariants of the system known as Casimir invariants. The relationships between the various actions are described, and it is shown that the leaf action is a link between actions written in terms of Lagrangian and Eulerian variables.