Abstract
The Landau-Lifshitz equation is considered as an approximation of the Abraham-Lorentz-Dirac equation. It is derived from the Abraham-Lorentz-Dirac equation by treating radiation reaction terms as a perturbation. However, while the Abraham-Lorentz-Dirac equation has pathological solutions of pre-acceleration and runaway, the Landau-Lifshitz equation and its finite higher order extensions are free of these problems. So it seems mysterious that the property of solutions of these two equations is so different. In this paper we show that the problems of pre-acceleration and runaway appear when one consider a series of all-order perturbation which we call it the Landau-Lifshitz series. We show that the Landau-Lifshitz series diverges in general. Hence a resummation is necessary to obtain a well-defined solution from the Landau-Lifshitz series. This resummation leads the pre-accelerating and the runaway solutions. The analysis is focusing on the non-relativistic case, but we can extend the results obtained here to relativistic case at least in one dimension.
Highlights
A charged particle emits radiation when it is accelerated
The ALD equation itself is well established but it has the infamous problem of runaway solutions which are solutions that describe the charged particle infinitely accelerated to the speed of light even when the external force vanishes
We find that the LL series diverges in general and is an asymptotic series of the solutions of ALD equation
Summary
A charged particle emits radiation when it is accelerated. Since the radiation carries energy and momentum, so the conservation laws require that the equation of motion for the charged particle should be modified by a friction term. The ALD equation itself is well established but it has the infamous problem of runaway solutions which are solutions that describe the charged particle infinitely accelerated to the speed of light even when the external force vanishes. When the external force vanishes, the backreaction vanishes, neither the pre-acceleration nor the runaway occurs in the LL equation This feature holds for the higher orders of the perturbation. Though the each term of LL series doesn’t have the problems of pre-acceleration and runaway, the resummation can lead these pathological solutions. We show that the LL series has the problems of pre-acceleration and runaway, in a way that quite similar to the case of the ALD equation.
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