Group of Canonical Diffeomorphisms and the Poisson-Vlasov Equations

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.