Delay Equations and Radiation Damping

TL;DR

This paper investigates how delay equations modeling field retardation effects lead to runaway modes in classical radiation reaction, identifying slow manifold solutions and bifurcation phenomena when retardation is large.

Contribution

It provides a new understanding of the origin of runaway modes and bifurcations in delay equations related to radiation damping, linking them to slow manifold dynamics.

Findings

Runaway modes are associated with solutions outside the slow manifold.

Small retardation effects lead to solutions on the slow manifold, avoiding runaway behavior.

Large retardation effects can cause bifurcations in the system's motion.

Abstract

Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.