Analytical solution of the Sommerfeld-Page equation

TL;DR

This paper presents an analytical solution to the Sommerfeld-Page equation, a delay differential equation modeling classical electron dynamics, inspired by recent solutions in epidemic modeling.

Contribution

It provides a pedagogical method to derive an analytical solution for the Sommerfeld-Page equation, filling a gap in the understanding of this classical physics problem.

Findings

Analytical solution derived for the Sommerfeld-Page equation.

Method inspired by recent epidemic modeling solutions.

Enhances understanding of delay differential equations in physics.

Abstract

The Sommerfeld-Page equation describes the non-relativistic dynamics of a classical electron modeled by a sphere of finite size with a uniform surface charge density. The Sommerfeld-Page equation is a delay differential equation, and almost no exact results on solutions of this equation was known. However, the progress has been made, especially in the last few years, and recently an analytical solution was found for an almost identical delay differential equation, which arose in the context of mathematical modeling of COVID-19 epidemics. Inspired by this research, we offer a pedagogical exposition of how one can find an analytical solution of the Sommerfeld-Page equation.