Newton like equations for the radiating particle: the non relativistic limit

TL;DR

This paper identifies conditions under which solutions to Newton-like and Abraham-Lorentz-Dirac equations coincide for radiating particles, avoiding unphysical behaviors, and introduces generalized functions to describe non-causal solutions in the non-relativistic limit.

Contribution

It demonstrates the existence of common solutions to both equations using generalized functions, and applies this to specific force scenarios, linking classical and extended particle models.

Findings

Common solutions exist for certain force classes.

Generalized functions describe non-causal, discontinuous motions.

Solutions match extended particle models in specific cases.

Abstract

A broad class of forces, P, is identified, for which the Abraham-Lorentz-Dirac (ALD) and Newton-like equations have solutions in common. Moreover, these solutions do not present pre-acceleration or escape into infinity (runaway behavior). Any continuous or piecewise continuous force can be represented in terms of functions belonging to this class P. It was also argued that the set of common solutions of both sets of equations is wider, it was proved that these solutions could be formulated in terms of generalized functions. The existence of such generalized functions motions is explicitly demonstrated for the relevant example of the instantaneously applied constant force, for which the respective solution of the ALD equation exhibits lack of causality and runaway motion. In this case, the expressions for the position and velocity of the particle are formulated in terms of generalized functions of time, having only a point support at the time that the force is applied. Thus, both, the velocity and the position are discontinuous at the instant that the force was applied. The solution for a particle moving between the plates of a capacitor reproduces the one obtained by A. Yaghjian from his discussion for extended particles. This outcome suggests a possible link or equivalence between both studies. A solution, common to the Newton-like and the ALD equations, for a constant homogeneous magnetic field is also presented. The extension of the results obtained to a relativistic context will be studied in future works.