On electromagnetic fields in the Hamiltonian description of continua

Abstract

The group theoretical structure of an infinite dimensional Hamiltonian formulation of continuum mechanics is studied using as an example the Maxwell-Vlasov system. In contrast to earlier works, electromagnetism and charged matter are coupled via Poisson brackets without using the vector potential. The charged matter is described on the group of canonical transformations on R 6 and we show that its evolution arises from a symplectic structure, modified by the magnetic field. The configurations of the electromagnetic field must be constrained by the physical requirement of the Gauss law. With the energy-functional taken as a Hamiltonian this leads - even for relativistic particles - to the well-known equations of motion.