Variational formulation of the electromagnetic radiation-reaction problem

TL;DR

This paper introduces a new variational approach to derive a consistent second-order radiation-reaction equation of motion for finite-size charged particles in classical electrodynamics, addressing issues with previous point-charge models.

Contribution

It proposes a novel variational formulation for the radiation-reaction problem using finite-size spherical-shell charges, resulting in a second-order differential equation consistent with fundamental physical principles.

Findings

Derives a second-order RR equation avoiding LAD's issues.

The new RR equation is valid even with sudden forces.

Recovers the LAD equation asymptotically.

Abstract

A fundamental issue in classical electrodynamics is represented by the search of the exact equation of motion for a classical charged particle under the action of its electromagnetic (EM) self-field - the so-called radiation-reaction equation of motion (RR equation). In the past, several attempts have been made assuming that the particle electric charge is localized point-wise (point-charge). These involve the search of possible so-called "regularization" approaches able to deal with the intrinsic divergences characterizing point-particle descriptions in classical electrodynamics. In this paper we intend to propose a new solution to this problem based on the adoption of a variational approach and the treatment of finite-size spherical-shell charges. The approach is based on three key elements: 1) the adoption of the relativistic synchronous Hamilton variational principle recently pointed out (Tessarotto et al, 2006); 2) the variational treatment of the EM self-field, for finite-size charges, taking into account the exact particle dynamics; 3) the adoption of the axioms of classical mechanics and electrodynamics. The new RR equation proposed in this paper, departing significantly from previous approaches, exhibits several interesting properties. In particular: a) unlike the LAD (Lorentz-Abraham-Dirac) equation, it recovers a second-order ordinary differential equation which is fully consistent with the law of inertia, Newton principle of determinacy and Einstein causality principle and b) unlike the LL (Landau-Lifschitz) equation, it holds also in the case of sudden forces. In addition, it is found that the new equation recovers the customary LAD equation in a suitable asymptotic approximation.