Radiation reaction in classical electrodynamics

The Lorentz-Abraham-Dirac (LAD) equations 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 bear several problems. In particular, almost all solutions are dynamically unstable, and therefore, highly questionable. As shown by Spohn et al., stable solutions to LAD equations can be approximated by means of singular perturbation theory in a certain regime and lead to the Landau-Lifshitz equation. However, for two charges them are also counterexamples, in which all solutions to LAD equations are unstable. The question remains whether better equations of motion than LAD equations can be found to describe the dynamics of charges in the electromagnetic fields. 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 equations 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's 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's 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 equations but provokes the mentioned dynamical instability by a mechanism we discuss and, as the derived equations of motion are explicit, is unnecessary.