Hamiltonian formulation of the Klein–Gordon–Maxwell equations

The nonlinear Klein-Gordon-Maxwell equations (NKGM) provide models for the\udinteraction between the electromagnetic field and matter. The relevance of NKGM relies on the fact\udthat they are the ‘‘simplest’’ gauge theory which is invariant under the group of Poincare´. These\udequations present the interesting phenomenon of solitons. In this paper, we show that NKGM present\udan Hamiltonian structure and hence they can be written as equations of the first order in t. This\udfact is not trivial since the Lagrangian does not depend on qtj (see section 3.3) and a suitable analysis\udof its structure is necessary. In the last section, we recall a recent result which states the existence\udof solitons by using the particular structure of the Hamiltonia