Relativistic Lagrangians for the Lorentz-Dirac equation

TL;DR

This paper introduces two novel relativistic Lagrangians formulated for the Lorentz-Dirac equation, one with an exponential damping function and another with auxiliary variables, both reproducing the equation with a source term.

Contribution

The work presents new relativistic Lagrangians for the Lorentz-Dirac equation, expanding the theoretical framework for describing charged particle dynamics.

Findings

Both Lagrangians produce the Lorentz-Dirac equation with a source-like term.

One Lagrangian explicitly depends on the world-line parameter, the other does not.

The formulations accommodate an arbitrary world-line parameter.

Abstract

We present two types of relativistic Lagrangians for the Lorentz-Dirac equation written in terms of an arbitrary world-line parameter. One of the Lagrangians contains an exponential damping function of the proper time and explicitly depends on the world-line parameter. Another Lagrangian includes additional cross-terms consisting of auxiliary dynamical variables and does not depend explicitly on the world-line parameter. We demonstrate that both the Lagrangians actually yield the Lorentz-Dirac equation with a source-like term.