An action principle for the Vlasov equation and associated Lie perturbation equations. Part 1. The Vlasov—Poisson system

A new action principle determining the dynamics of the Vlasov-Poisson system is presented (the Vlasov-Maxwell system will be considered in Part 2). The particle distribution function is explicitly a field to be varied in the action principle, in which only fundamentally Eulerian variables and fields appear. The Euler-Lagrange equations contain not only the Vlasov-Poisson system but also equations associated with a Lie perturbation calculation on the Vlasov equation. These equations greatly simplify the extensive algebra in the small-amplitude expansion. As an example, a general, manifestly Manley-Rowesymmetric, expression for resonant three-wave interaction is derived. The new action principle seems ideally suited for the derivation of action principles for reduced dynamics by the use of various averaging transformations (such as guiding-centre, oscillation-centre or gyro-centre transformations). It is also a powerful starting point for the application of field-theoretical methods. For example, the recently found Hermitian structure of the linearized equations is given a very simple and instructive derivation, and so is the well-known Hamiltonian bracket structure of the Vlasov-Poisson system.