Adiabatic Limit for the Maxwell{Lorentz Equations

Markus Kunze1 Mathematisches Institut der Universit at K oln Weyertal 86, D-50931 K oln, Germany email: [email protected] Herbert Spohn Zentrum Mathematik and Physik Department, TU M unchen D-80290 M unchen, Germany email: [email protected] Abstract We consider the Abraham model of a rigid charge distribution coupled to the electromagnetic eld 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 e ective 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 elds are derived from the Li enard{Wiechert potentials of a moving point charge and that the radiated energy is given by Larmor's formula.