Newton-like equations for a radiating particle

TL;DR

This paper introduces Newton-like equations for a radiating particle that align with the Abraham-Lorentz-Dirac equations, avoiding runaway solutions and pre-acceleration, supported by numerical analysis and generalized modifications.

Contribution

It presents a new form of Newton equations for radiating particles that satisfy ALD equations and addresses issues like runaway solutions through smoothness conditions and effective forces.

Findings

Newton-like equations match ALD equations for general motions.

Numerical solutions confirm the absence of runaway and pre-acceleration effects.

Modified ALD equations handle discontinuous forces and support instantaneous force vanishing.

Abstract

Second order Newton equations of motion for a radiating particle are presented. It is argued that the trajectories obeying them also satisfy the Abraham-Lorentz-Dirac (ALD) equations for general 3D motions in the non-relativistic and relativistic limits. The case of forces only depending of the proper time is here considered. For these properties to hold, it is sufficient that the external force to be infinitely smooth and that a series formed with its time derivatives converges. This series define in a special local way the effective forces entering the Newton equations. When the external force vanishes in an open vicinity of a given time, the effective one also becomes null. Thus, the proper solutions of the effective equations can not show runaway or pre-acceleration effects. The Newton equations are numerically solved for a pulsed force given by an analytic function along the proper time axis. The simultaneous satisfaction of the ALD equations is numerically checked. Further, a set of modified ALD equations for almost everywhere infinitely smooth forces, but including step like discontinuities in some points is also presented for the case of collinear motions. The form of the equations supports the statement argued in a previous work, about that the causal Lienard-Wiechert field solution surrounding a radiating particle, implies that the effective force on the particle should instantaneously vanish when the external force is retired. The modified ALD equations proposed in the previous work are here derived in a generalized way including the same effect also when a the force is instantly connected.