# A non-perturbative proof of Bertrand's theorem

## TL;DR

This paper presents a non-perturbative proof of Bertrand's theorem, demonstrating that only the Newtonian and isotropic harmonic oscillator potentials produce closed orbits.

## Contribution

It offers a concise, non-perturbative proof identifying the only two central potentials with closed orbits, simplifying previous approaches.

## Key findings

- Confirmed only Newtonian and harmonic oscillator potentials produce closed orbits.
- Provided a direct, non-perturbative proof of Bertrand's theorem.
- Simplified understanding of conditions for closed orbits in central potentials.

## Abstract

We discuss an alternative non-perturbative proof of Bertrand's theorem that leads in a concise way directly to the two allowed fields: the newtonian and the isotropic harmonic oscillator central fields.