A relativistic analogue of the Kepler problem

Abstract

The Poincaré invariant system of two point particles with an instantaneous interaction-at-a-distance originally proposed by Fokker is studied in the Hamiltonian formalism. The interaction, which agrees to first order in the coupling constant with the electromagnetic one obtained from the Liénard-Wiechert fields, is described in an advanced-retarded state space. The first particle moves in the advanced field of the second which in turn is subject to the retarded field of the first. The acceleration terms in the Liénard-Wiechert fields are neglected.

In this theory the state space of the system is a twelve-dimensional manifold Σ and the motions are described as integral curves of a vector field that is obtained as the projection of the generator of time translations in space-time. The Poincaré group acts on this manifold Σ in a well-defined way and leaves a symplectic form ω invariant. Thus the set of all possible motions of this system can be studied by the methods of modern symplectic mechanics. In this paper the general method is explained and the set of all bounded motions for two equal rest masses and an attractive force is studied qualitatively and numerically. In the limit (binding energy)/(sum of rest masses) · (speed of light)² → 0 all the features of the classical Kepler motion are obtained.