Inverse problem and Bertrand's theorem

TL;DR

This paper reformulates Bertrand's theorem as an inverse problem in classical mechanics, providing a novel, elegant proof by solving a numerical equation within a restricted class of solutions.

Contribution

It introduces a new inverse problem approach to Bertrand's theorem, enabling a compact proof through numerical equation solutions.

Findings

Solutions can be obtained by solving a numerical equation within a restricted class.

Provides a compact and elegant proof of Bertrand's theorem.

Reformulates classical problem as an inverse problem.

Abstract

The Bertrand's theorem can be formulated as the solution of an inverse problem for a classical unidimensional motion. We show that the solutions of these problems, if restricted to a given class, can be obtained by solving a numerical equation. This permit a particulary compact and elegant proof of Bertrand's theorem.