The Electromagnetic Three-Body Problem With Radiation Terms – Existence-Uniqueness of Periodic Orbit (II)

Abstract

The primary goal of the present paper is to prove an existence-uniqueness of periodic solution of the equations of motion for the 3-body problem of classical electrodynamics. The equations of motion were derived in a recent paper of the author. Particular case of this problem is the He-atom – the simplest multi-electronic atom. We have applied our previous results to 3-body problem introducing radiation terms and in this manner we have obtained a system of 12 equations of motion. We have proved that three equations are a
consequence of the first 9 ones, so that we consider 9 equations for 9 unknown functions. We introduce a suitable operator in a specific function space and formulate conditions for the existence-uniqueness of fixed point of this operator that is a periodic solution of the 3-body equations of motion. Finally, we verify the conditions obtained for the He-atom.