Slow Motion of Charges Interacting Through the Maxwell Field

TL;DR

This paper analyzes the slow motion of multiple charges interacting via the Maxwell field, demonstrating that over long timescales their dynamics are well approximated by the Darwin Lagrangian, with detailed consideration of electromagnetic effects.

Contribution

The paper rigorously derives the Darwin Lagrangian as an effective description for charges interacting through the Maxwell field over extended timescales, including mass renormalization and electromagnetic corrections.

Findings

Charges follow the Darwin Lagrangian dynamics over long times.

Coulomb forces dominate interactions at order ^2.

Magnetic and retardation effects contribute to Darwin correction.

Abstract

We study the Abraham model for $N$ charges interacting with the Maxwell field. On the scale of the charge diameter, $R_{\phi}$, the charges are a distance $\eps^{-1}R_{\phi}$ apart and have a velocity $\sqrt{\eps} c$ with $\eps$ a small dimensionless parameter. We follow the motion of the charges over times of the order $\eps^{-3/2}R_{\phi}/c$ and prove that on this time scale their motion is well approximated by the Darwin Lagrangian. The mass is renormalized. The interaction is dominated by the instantaneous Coulomb forces, which are of the order $\eps^{2}$. The magnetic fields and first order retardation generate the Darwin correction of the order $\eps^{3}$. Radiation damping would be of the order $\eps^{7/2}$.