Conservation of Momentum and Energy in the Lorenz-Abraham-Dirac Equation of Motion

TL;DR

This paper examines the conditions under which the Lorentz-Abraham-Dirac equations conserve momentum and energy, analyzing effects of mass renormalization and comparing solutions for a charge in a capacitor.

Contribution

It provides a concise derivation of conservation conditions for modified LAD equations and compares various solutions in a specific physical scenario.

Findings

Conservation of momentum and energy depends on specific velocity and force conditions.

Mass renormalization influences the radiated momentum-energy.

Solutions for different equations are obtained for a charge in a parallel-plate capacitor.

Abstract

After a brief review of the modified (by transition forces) causal Lorentz-Abraham (LA) classical equation of motion for an extended charged sphere and its limit to the mass-renormalized modified causal Lorentz-Abraham-Dirac (LAD) equation of motion as the radius of the charged sphere approaches zero, a concise derivation is given for the conditions on the velocity and external force required for these modified equations of motion to satisfy conservation of momentum and energy. The effects of mass renomalization on the radiated momentum-energy is clarified. The solutions to the unmodified LAD equation of motion, the causal modified LA and LAD equations of motion, and the Landau-Lifshitz approximate solution to the unmodified LAD equation of motion are obtained for a charge traveling through a parallel-plate capacitor.