Asymptotics of a proposed delay-differential equation of motion for charged particles

Abstract

We study the behavior in the remote past and future of solutions of an equation of motion for charged particles proposed by F. Rohrlich, for the special case in which the motion is in one spatial dimension. We show that if an external force is applied for a finite time, some solutions exhibit the property of ``preacceleration'', meaning that the particle accelerates before the force is applied, but that there do exist solutions without preacceleration. However, most solutions without preacceleration exhibit ``postacceleration'' into the infinite future (i.e., the particle accelerates after the force is removed). Some may consider such behavior as sufficiently "unphysical" to rule out the equation. More encouragingly, we show that analogs of the unphysical ``runaway'' solutions of the Lorentz-Dirac equation do not occur for solutions of Rohrlich's equation. We show that when the external force eventually vanishes, the proper acceleration vanishes asymptotically in the future, and the coordinate velocity becomes asymptotically constant.