On the Zoll deformations of the Kepler problem

TL;DR

This paper extends classical results by showing an infinite-dimensional family of Zoll potentials on the plane, where all non-collision orbits at a fixed energy are closed, and explores their properties and invariances.

Contribution

It introduces the concept of Zoll deformations for the Kepler problem, revealing a rich structure of central potentials with closed orbits beyond classical cases.

Findings

Existence of infinite-dimensional space of Zoll potentials at a fixed energy

Classification of natural rotationally invariant Zoll systems on the plane

Rigidity results for systems Zoll at multiple energies

Abstract

A celebrated result of Bertrand states that the only central force potentials on the plane with the property that all bounded orbits are periodic are the Kepler potential and the potential of the harmonic oscillator. In this paper, we complement Bertrand's theorem showing the existence of an infinite dimensional space of central force potentials on the plane which are Zoll at a given energy level, meaning that all non-collision orbits with given energy are closed and of the same length. We also determine all natural systems on the (not necessarily flat) plane which are invariant under rotations and Zoll at a given energy and prove several existence and rigidity results for systems which are Zoll at multiple energies.