Asymptotic character of the series of classical electrodynamics and an application to bremsstrahlung

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 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 e2/c. Finally, we discuss the implications for the radiated spectrum, showing that the threshold in angular momentum should correspond to a frequency cutoff of the order of magnitude of the de Broglie frequency.