Possible potentials responsible for stable circular relativistic orbits

TL;DR

This paper extends Bertrand's theorem to relativistic central force systems, deriving stability criteria and analyzing which potentials can produce stable, circular orbits under relativistic conditions.

Contribution

It generalizes Bertrand's theorem to relativistic systems and provides a stability criterion for potentials capable of producing stable circular orbits.

Findings

Inverse square law remains stable under relativistic conditions

Simple harmonic motion potential is unstable relativistically

Relativistic effects prevent stable orbits with linear restoring forces

Abstract

Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In the present article an attempt has been made to generalize Bertrand's theorem to the central force problem of relativistic systems. The stability criterion for potentials which can produce stable, circular orbits in the relativistic central force problem has been deduced and a general solution of it is presented in the article. It is seen that the inverse square law passes the relativistic test but the kind of force required for simple harmonic motion does not. Special relativistic effects do not allow stable, circular orbits in presence of a force which is proportional to the negative of the displacement of the particle from the potential center.