The determination of the apsidal angles and Bertrand's theorem

TL;DR

This paper derives a general expression for apsidal angles in arbitrary central potentials and provides a new proof of Bertrand's theorem by analyzing conditions for energy and angular momentum independence.

Contribution

It introduces a universal formula for apsidal angles and offers a non-perturbative proof of Bertrand's theorem based on these conditions.

Findings

Apsidal angles can be determined for any central potential.

Conditions for energy and angular momentum independence are identified.

A new proof of Bertrand's theorem is established.

Abstract

We derive an expression for the determination of the apsidal angles that holds good for arbitrary central potentials. Then we discuss under what conditions the apsidal angles remain independent of the mechanical energy and angular momentum in the central force problem. As a consequence, an alternative and non-perturbative proof of Bertrand's theorem is obtained.