Geometry of plasma dynamics. I. Group of canonical diffeomorphisms

The dynamics of collisionless plasma described by the Poisson-Vlasov equations is connected to the Hamiltonian motions of particles and their symmetries. The group of canonical diffeomorphisms of particle phase space is described and adopted as the configuration space. The dual space of its Lie algebra of Hamiltonian vector fields is identified with the space of nonclosed one-forms. 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. Usual 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 associated with the action of additive group of functions of particle phase space. Equivalence of Hamiltonian functionals in momentum and density formulations is shown. As an alternative formulation in momentum variables, a canonical Hamiltonian system with a quadratic Hamiltonian functional is described. In the case that the groups of Hamiltonian and volume preserving diffeomorphisms coincide, a comparison of one-dimensional plasma and two-dimensional incompressible fluid is presented.