Classical Helium Atom with Radiation Reaction

TL;DR

This paper investigates a classical Helium atom model incorporating radiation reaction forces, revealing new qualitative features, resolving inconsistencies, and analyzing the atom's formation process through numerical studies of invariant measures.

Contribution

It introduces a classical Helium atom model with radiation reaction, applying Dirac's prescription to eliminate unphysical solutions and studying the atom's formation via invariant measures.

Findings

Dirac's prescription resolves model inconsistencies.

Most initial data lead to atom breakdown without radiation reaction.

Invariant measure on the zero-dipole manifold may be singular.

Abstract

We study a classical model of Helium atom in which, in addition to the Coulomb forces, the radiation reaction forces are taken into account. This modification brings in the model a new qualitative feature of a global character. Indeed, as pointed out by Dirac, in any model of classical electrodynamics of point particles involving radiation reaction one has to eliminate, from the a priori conceivable solutions of the problem, those corresponding to the emission of an infinite amount of energy. We show that the Dirac prescription solves a problem of inconsistency plaguing all available models which neglect radiation reaction, namely, the fact that in all such models most initial data lead to a spontaneous breakdown of the atom. A further modification is that the system thus acquires a peculiar form of dissipation. In particular, this makes attractive an invariant manifold of special physical interest, the zero--dipole manifold, that corresponds to motions in which no energy is radiated away (in the dipole approximation). We finally study numerically the invariant measure naturally induced by the time--evolution on such a manifold, and this corresponds to studying the formation process of the atom. Indications are given that such a measure may be singular with respect to that of Lebesgue.