Hamiltonian structure for classical electrodynamics of a point particle

Abstract

We prove that, contrary to the common belief, the classical Maxwell electrodynamics of point-like particle may be formulated as an infinite dimensional Hamiltonian system. We derive a well defined quasi-local Hamiltonian which possesses direct physical interpretation being equal to the total energy of the composed system (field + particle). The phase space of this system is endowed with a very interesting symplectic structure. We prove that this structure is strongly nondegenerate and, therefore, enables one to define the consistent Poisson bracket for the particle and field degrees of freedom. We stress that this formulation is perfectly gauge invariant.