Hamiltonian formulation of the Klein-Gordon-Maxwell equations (Q634426)

The nonlinear Klein-Gordon-Maxwell equations (NKGM) provide models for the interaction between the electromagnetic field and matter. The relevance of NKGM relies on the fact that they are the ``simplest'' gauge theory which is invariant under the group of Poincaré. These equations present the interesting phenomenon of solitons. In this article, the writers show that NKGM present an Hamiltonian structure and hence they can be written as equations of the first order in \(t\). This fact is not trivial since the Lagrangian does not depend on \(\partial_{t}\varphi\) (see sec. 3.3) and a suitable analysis of its structure is necessary. In the last section, the authors recollect a recent result which states the existence of solitons by using the particular structure of the Hamiltonian.