Adiabatic limit for the Maxwell-Lorentz equations

Abstract

We consider the Abraham model of a rigid charge distribution coupled to the electromagnetic field and subject to, on the scale of the charge diameter, slowly varying external potentials. We prove that in the adiabatic limit the motion of the charged particle is governed by a effective Hamiltonian. To next order one has to add as a small correction the relativistically covariant radiation reaction. This third order equation has a repulsive center manifold and the true solution is well approximated by a trajectory on the center manifold. We also prove that in the adiabatic limit the fields are derived from the Liénard-Wiechert potentials of a moving point charge and that the radiated energy is given by Larmor's formula.