A remark on rational isochronous potentials

TL;DR

This paper classifies rational isochronous potentials in one-dimensional mechanical systems, showing they are either quadratic or quadratic with an inverse-square term, up to shifts and constants.

Contribution

It provides a complete characterization of rational isochronous potentials, identifying their explicit forms and conditions for periodic solutions.

Findings

Rational isochronous potentials are either quadratic or quadratic plus inverse-square terms.

All such potentials are equivalent to these forms up to shifts and additive constants.

The classification simplifies understanding of periodic solutions in these systems.

Abstract

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form $U(x) = 1/2 \omega^2 x^2$ or $U(x) = 1/8 \omega^2 x ^2 + c^2 x^{-2}.$