The effective analogous potential energy function (EAPEF): a tool for analyzing motion on a surface of revolution

We show that the motion of a particle sliding along a frictionless surface of revolution z(r) and subject to a uniform gravitational field can be reduced to a one-dimensional problem of radial motion with an effective analogous potential energy function (EAPEF). The EAPEF depends on the initial angular momentum and total energy of the particle, and therefore depends on the initial conditions of the motion. A given surface z(r) produces a family of different EAPEFs. We derive the general equation for the EAPEF and find the radial equation of motion, the conditions for circular orbits on the surface, and the stability of circular orbits under small radial oscillations. The EAPEF can be used to find the period of radial oscillations in a perturbed circular orbit. Comparing the periods of the circular orbit and small radial oscillations, we determine the initial conditions with which the two periods will be in resonance and create a closed orbit. Our results are applied to the case of a conic surface. Simulations illustrating these results are available at osp.berry.edu/surfaces/surfaces.xhtml.