Group of Canonical Diffeomorphisms and the Poisson-Vlasov Equations

TL;DR

This paper explores the geometric and Hamiltonian structures underlying the Poisson-Vlasov equations in plasma physics, connecting symmetries, reduction techniques, and alternative formulations.

Contribution

It introduces a new momentum map perspective and compares plasma dynamics with fluid models, advancing the geometric understanding of collisionless plasma.

Findings

Derivation of the Poisson equation from gauge symmetries.

Reduced dynamical equations via variational derivatives.

Comparison between plasma and fluid dynamics models.

Abstract

Dynamics of collisionless plasma described by the Poisson-Vlasov equations is connected with the Hamiltonian motions of particles and their symmetries. The Poisson equation is obtained as a constraint arising from the gauge symmetries of particle dynamics. Variational derivative constrained by the Poisson equation is used to obtain reduced dynamical equations. Lie-Poisson reduction for the group of canonical diffeomorphisms gives the momentum-Vlasov equations. Plasma density is defined as the divergence of symplectic dual of momentum variables. This definition is also given a momentum map description. An alternative formulation in momentum variables as a canonical Hamiltonian system with a quadratic Hamiltonian functional is described. A comparison of one-dimensional plasma and two-dimensional incompressible fluid is presented.