Nonlinearity zyxwvutsr 6 zyxwvutsr (1993) 905-914. Printed in the UK Asymptotic character of the series of classical electrodynamics and an application to bremsstrahlung Andrea Caratit and Luigi Galganil: t Universita di Padova, Dipartimento di Matematica Pura e Applicata Via Belzoni 7. 35121 Padova, Italy i Universiti di Milano, Dipartimento di Matematica Via Saldini 50, 20133 Milano, Italy Received 15 February 1993 Recommended by P Goddard Absbad. The Lorentz-Dirac equation. which describes the self-interaction of a classical charged particle with the electromagnetic field, is studied, for the case of scattering and in the non-relativistic approximation, in the framework of the theory of singular perturbation problems. We prove that the series expansions, which are usually given for the solutions in terms of the electric charge, in general are divergent and have asymptotic character. A closer inspection of such series leads to recognition of two types of particle motions, namely those qualitatively similar to purely mechanical zy ones (corresponding to vanishing charge), and those qualitatively dissimilar. For an attractive Coulomb potential, the distinction turns out to depend on the value of the initial angular momentum, the threshold being of the order of magnitude of e'lc. Finally, we discuss the implications for the radiated spectrum, showing that the threshold in angular momentum should correspond to a frequency zyxwv cutoff of the order of magnitude of the de Broglie frequency. PAC numbers: 0350,0365,0230 1. Introduction It is very well known that the series expansions occurring in quantum electrodynam- ics are in general expected to be divergent and (at most) of asymptotic type (see [l], [2] page 84, and [3] chapter 37), and it thus seems quite natural to ask whether the same characteristics are shared by the series expansions of classical electrodynamics, typically those obtained from the Lorentz-Dirac equation (see for example zy [4], section 17.6). However, to our surprise. we couldn't find any clear statement on the subject, and even met with statements suggesting a possible analytic character. Indeed, in the concluding section of a review article by a well respected scientist (see [SI page 362, and also zyxwvut [6]) one finds the following sentence: 'The existence of solutions zyxwvut of the Lorentz-Dirac equation was proven. But their uniqueness has not been proven. zyxwvuts Nor is 'the dependence of these solutions 'on the charge e known. Are 0951-7715/93/060905 + 10$U7.50 zyxwvuts 0 1993 IOP Publishing Ltd and LMS Publishing Ltd 905