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

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 runaway solutions. Runaways violate conservation of energy while preacceleration violates causality. In this work, I will focus on one of the most used alternative equations of motion: the Landau-Lifshitz equation, which has no pathological solution. However, it is a first-order approximation to the Abraham-Lorentz equation, raising the question of how an approximation turns out to be better than the original. Finally, some numerical results for a variety of external forces are presented to compare both the equations.