#### Document Type

Article

#### Publication Date

2008

#### Department

Mathematics

#### 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.

#### Recommended Citation

Santoprete, Manuele, "Gravitational and Harmonic Oscillator Potentials on Surfaces of Revolution" (2008). *Mathematics Faculty Publications*. 44.

http://scholars.wlu.ca/math_faculty/44

## Comments

This article was originally published in

Journal of Mathematical Physics49: 042903. © 2008 American Institute of Physics.