Poisson bracket for the Vlasov equation on a symplectic leaf

Abstract

The degeneracy in the Lie-Poisson bracket, associated with the Hamiltonian structure of the Vlasov equation, is removed by restriction to a given symplectic leaf. The restricted equation of motion in terms of a generating function and it manifestly preserves the Casimir constraints of the system. A nondegenerate Poisson bracket in terms of the generating function is presented.