Runaway charged particles and center manifolds.pdf

Runaway Charged Particles and Center ManifoldsHerbert SpohnZentrum Mathematik, TU Munchen, D{80290 Munchen, Germany(August 26, 1998)AbstractWe consider the coupled Maxwell{Lorentz equations for a single charge witha rigid charge distribution subject to slowly varying external potentials. Inthe adiabatic limit, to lowest order, the motion of the charge is governed byan eective Hamiltonian. In the next order we obtain, as a small correction,the radiation reaction containing


q{terms. This equation has a stable centermanifold and the true solution stays close to the center manifold. The motionon the center manifold is governed by a novel second order equation whichcontains \friction terms whose sign and strength depend on the externalpotentials. Typeset using REVTEX1 Classical electrodynamics covers a huge range of physical phenomena. One considerseither the motion of charged particles in prescribed electromagnetic elds or the electro-magnetic elds for given world lines of the charges. With Abraham and Lorentz it is thennatural to assume that also the coupled system, elds and charges, is a well{dened theory.Consulting the standard textbook [1] leaves the reader unsatised, a feeling reinforced by therather gloomy discussion of Feynman [2]. The problem is centered around the point chargelimit. The electromagnetic contribution to the mass of the particle tends then to +1 whichis supposedly balanced by taking its bare mass to 1. The resulting eective equationsof motion for the charged particle are of third order and contain a


q{term, which leads torunaway solutions: the mechanical energy of the particle increases at the expense of theinnite self{energy. Rohrlich [3] rules such solutions as unphysical, hence to be ignored. Hisprescription seems somewhat ad hoc and in addition leads to the acausal preacceleration. Irefer to the book by Yaghjian [4] as an excellent and comprehensive survey.Recently we investigated the Maxwell{Lorentz equations for a single cha