# On Radiation Reaction in Classical Electrodynamics

## TL;DR

This paper proposes new equations of motion for classical charged particles that address the instabilities of the Lorentz-Abraham-Dirac equations by deriving delay differential equations based on a more justified application of electromagnetic theory.

## Contribution

It introduces a novel derivation method for equations of motion that avoids LAD's instabilities by using an extended charge model and delay differential equations.

## Key findings

- Derived second order delay differential equations for radiation reaction
- Revealed the instability mechanism in LAD through Taylor expansion analysis
- Provided explicit equations of motion avoiding LAD's problematic solutions

## Abstract

The Lorentz-Abraham-Dirac equations (LAD) may be the most commonly accepted equation describing the motion of a classical charged particle in its electromagnetic field. However, it is well known that they bare several problems. In particular, almost all solutions are dynamically unstable, and therefore, highly questionable. The question remains whether better equations of motion than LAD can be found to describe the dynamics of charges in the electromagnetic fields. In this paper we present an approach to derive such equations of motions, taking as input the Maxwell equations and a particular charge model only, similar to the model suggested by Dirac in his original derivation of LAD in 1938. We present a candidate for new equations of motion for the case of a single charge. Our approach is motivated by the observation that Dirac's derivation relies on an unjustified application of Stokes' theorem and an equally unjustified Taylor expansion of terms in his evolution equations. For this purpose, Dirac's calculation is repeated using an extended charge model that does allow for the application of Stokes' theorem and enables us to find an explicit equation of motion by adapting Parrott's derivation, thus avoiding a Taylor expansion. The result are second order differential delay equations, which describe the radiation reaction force for the charge model at hand. Their informal Taylor expansion in the radius of the charge model used in the paper reveals again the famous triple dot term of LAD but provokes the mentioned dynamical instability by a mechanism we discuss and, as the derived equations of motion are explicit, is unnecessary.