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

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.

💡 Research Summary

The paper presents an exact analytical solution of the non‑relativistic Landau‑Lifshitz (LL) equations for a point charge moving in a static Coulomb potential while radiating. Starting from the Lorentz‑Abraham‑Dirac (LAD) equation, the authors adopt the LL reduction, which replaces the problematic third‑order time derivative with a first‑order radiation‑reaction term that is accurate to order (q^{2}) and avoids runaway solutions. In the Coulomb field (V(r)=-k/r) (with (k>0)), the motion is confined to a plane, allowing the use of polar coordinates ((r,\theta)).

Conservation of angular momentum is broken by radiation damping; the LL equation yields a simple differential law for the angular momentum (L(t)):
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