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

Gravitational and harmonic oscillator potentials on surfaces of revolution