Relativistic effects in the dynamics of a particle in a Coulomb field

TL;DR

This paper investigates the impact of relativistic effects on a particle's motion in a Coulomb field, demonstrating the absence of Bertrand's property and analyzing integrability and collision behaviors.

Contribution

It provides new insights into the relativistic Coulomb problem, showing that classical properties like Bertrand's do not hold and exploring integrability and asymptotic collision behaviors.

Findings

Bertrand's property cannot occur in relativistic Coulomb systems

Analysis of the period function reveals differences from classical cases

Collision solutions exhibit specific asymptotic behaviors

Abstract

We prove that Bertrand's property cannot occur in a special-relativistic scenario using the properties of the period function of planar centres. We also explore some integrability properties of the relativistic Coulomb problem and the asymptotic behavior of collision solutions.