Abstract
The Landau-Lifshitz equation is the first in an infinite series of approximations to the Lorentz-Abraham-Dirac equation obtained from `reduction of order'. We show that this series is divergent, predicting wildly different dynamics at successive perturbative orders. Iterating reduction of order ad infinitum in a constant crossed field, we obtain an equation of motion which is free of the erratic behaviour of perturbation theory. We show that Borel-Pad\'e resummation of the divergent series accurately reproduces the dynamics of this equation, using as little as two perturbative coefficients. Comparing with the Lorentz-Abraham-Dirac equation, our results show that for large times the optimal order of truncation typically amounts to using the Landau-Lifshitz equation, but that this fails to capture the resummed dynamics over short times.
Highlights
Radiation reaction (RR) in electrodynamics becomes relevant in the presence of strong fields, where RR forces can become comparable to, or dominate, the Lorentz force
We focus mainly on the case of a constant crossed field (CCF) background, which is relevant to the Ritus-Narozhny conjecture on the breakdown of perturbation theory in strong fields [34]
We find that infinite reduction of order gives results which are free of the erratic behavior seen in perturbation theory, and which match LAD, but without the nonperturbative, and unphysical runaways or preacceleration behavior
Summary
Radiation reaction (RR) in electrodynamics becomes relevant in the presence of strong fields, where RR forces can become comparable to, or dominate, the Lorentz force. Vanishing final acceleration [11], one has an initialboundary value problem, and solutions (the existence and uniqueness of which is not guaranteed [19,20,21,22]) exhibit preacceleration before the external field is encountered (albeit on a small timescale of about 2 fm=c) Both analytical and numerical solutions of LAD are hampered by the strong nonlinearities present in the fully relativistic case [23]. IV we use the convergent strong-field behavior to improve our perturbative resummation, and obtain an analytical expression for LL∞ We compare this with the numerical solution of LAD.
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