Reduction of Order and Transseries Structure of Radiation Reaction

TL;DR

This paper analyzes the reduction of order in radiation reaction equations, revealing the transseries structure and non-perturbative effects that influence runaway solutions, using Borel plane and transseries techniques.

Contribution

It provides a detailed transseries and Borel plane analysis of the Landau-Lifshitz and Lorentz-Abraham-Dirac equations, highlighting non-perturbative aspects of reduction of order.

Findings

Reduction of order eliminates runaway solutions after resummation.

Non-perturbative formulation can retain runaway solutions.

Transseries analysis reveals complex solution structures under variable changes.

Abstract

The Landau-Lifshitz equation is obtained from the Lorentz-Abraham-Dirac equation through `reduction of order'. It is the first in a divergent series of approximations that, after resummation, eliminate runaway solutions. Using Borel plane and transseries analysis we explain why this is, and show that a non-perturbative formulation of reduction of order can retain runaway solutions. We also apply transseries analysis to solutions of the Lorentz-Abraham-Dirac equation, essentially treating them as expansions in both time and a coupling. Our results illustrate some aspects of such expansions under changes of variables and limits.