  We consider the nonlinear Klein Gordon Maxwell system on four dimensional
Minkowski space-time. For appropriate nonlinearities the system admits soliton
solutions which are gauge invariant generalizations of the non-topological
solitons introduced and studied by T.D. Lee and collaborators for pure complex
scalar fields. We develop a rigorous dynamical perturbation theory for these
solitons in the small e limit, where e is the electromagnetic coupling
constant. The main theorems assert the long time stability of the solitons with
respect to perturbation by an external electromagnetic field produced by the
background current, and compute their effective dynamics to O(e). The effective
dynamical equation is the equation of motion for a relativistic particle acted
on by the Lorentz force law familiar from classical electrodynamics.
