Elimination of Pathological Solutions of the Abraham-Lorentz Equation of Motion

Abstract

For more than a century the Abraham-Lorentz equation has generally been regarded as the correct description of the dynamics of a charged particle. However, there are pathological solutions of the Abraham-Lorentz equation in which a particle accelerates in advance of the application of a force, the so-called preacceleration solutions, and solutions in which the particle spontaneously accelerates even in the absence of an external force, also known as the runaway solutions. Runaways violate conservation of energy, and preacceleration violates causality. In this study, I shall focus on one of the most used alternative equations of motion: the Landau-Lifshitz equation, which has no pathological solutions. However, it is a first-order approximation to the Abraham-Lorentz equation, raising the question of how an approximation can be considered more accurate than the original.Finally, I shall present some numerical results for a variety of external forces and compare both the equations.