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We consider the nonlinear Klein Gordon Maxwell system on four dimensionalMinkowski space-time. For appropriate nonlinearities the system admits solitonsolutions which are gauge invariant generalizations of the non-topologicalsolitons introduced and studied by T.D. Lee and collaborators for pure complexscalar fields. We develop a rigorous dynamical perturbation theory for thesesolitons in the small e limit, where e is the electromagnetic couplingconstant. The main theorems assert the long time stability of the solitons withrespect to perturbation by an external electromagnetic field produced by thebackground current, and compute their effective dynamics to O(e). The effectivedynamical equation is the equation of motion for a relativistic particle actedon by the Lorentz force law familiar from classical electrodynamics.

Effective dynamics for solitons in the nonlinear Klein Gordon Maxwell system and the Lorentz force law