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

TL;DR

This paper examines the Abraham-Lorentz equation's pathological solutions, discusses how the Landau-Lifshitz equation avoids these issues, and compares their numerical solutions under various external forces.

Contribution

It analyzes the elimination of runaway and preacceleration solutions by using the Landau-Lifshitz equation as an improved alternative.

Findings

Landau-Lifshitz equation has no pathological solutions

Numerical comparisons show differences under various external forces

Landau-Lifshitz approximates Abraham-Lorentz better in certain regimes

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 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.