Exact Solution of the Landau-Lifshitz Equations for a Radiating Charged Particle in The Coulomb Potential

TL;DR

This paper provides an exact analytical solution to the classical Landau-Lifshitz equations for a charged particle in a Coulomb potential, incorporating radiation damping, and demonstrates the particle's spiral into the nucleus.

Contribution

It presents the first exact solution involving Painleve transcendent for a charged particle with radiation damping in a Coulomb field.

Findings

Negatively charged particle spirals into the nucleus

Supports the validity of the Landau-Lifshitz equation

Solution involves Painleve transcendent of type II

Abstract

We solve exactly the classical non-relativistic Landau-Lifshitz equations of motion for a charged particle moving in a Coulomb potential, including radiation damping. The general solution involves the Painleve transcendent of type II. It confirms our physical intuition that a negatively charged classical particle will spiral into the nucleus, supporting the the validity of the Landau-Lifshitz equation.