Gravitational and harmonic oscillator potentials on surfaces of revolution

TL;DR

This paper investigates particle motion on surfaces of revolution under central forces, identifying specific potentials that produce closed orbits, generalizing classical results like Bertrand's theorem to curved surfaces.

Contribution

It proves the existence of at most two special potentials for closed orbits on such surfaces and generalizes Bertrand's theorem to surfaces with constant Gaussian curvature.

Findings

Exactly two potentials produce closed orbits on some surfaces of constant curvature.

Surfaces may admit only one potential for closed orbits, which is a generalized harmonic oscillator.

A generalized Bertrand theorem is established for surfaces of revolution with constant Gaussian curvature.

Abstract

In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are closed and that there are exactly two on some surfaces with constant Gaussian curvature. The two potentials leading to closed orbits are suitable generalizations of the gravitational and harmonic oscillator potential. We also show that there could be surfaces admitting only one potential that leads to closed orbits. In this case, the potential is a generalized harmonic oscillator. In the special case of surfaces of revolution with constant Gaussian curvature, we prove a generalization of the well-known Bertrand theorem.