  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.
