Analyzing motion on a surface of revolution using the effective analogous potential energy function

The motion of a particle sliding on a frictionless surface of revolution z(r) subject to a uniform gravitational field can be reduced to a one-dimensional radial motion governed by an “effective analogous potential energy function” (EAPEF), similar to the effective potential energy function used in two-dimensional problems with radial symmetry. The EAPEF depends on the angular momentum and total energy of the particle, and thus on the initial conditions of the motion. The general expression for the EAPEF is derived and the EAPEF is used to find the radial equation of motion, the condition for circular orbits on the surface, and the period and stability of those orbits. For stable circular orbits, the EAPEF is used to find the period of small radial oscillations in a perturbed but stable circular orbit. By comparing the period of the circular orbit with the period of small radial oscillations, the conditions for which the perturbed orbit will be periodic are derived. The period of more general radial oscillations, as well as the average orbital period, is given in terms of integrals involving the EAPEF, which can be computed numerically and used to find periodic orbits with large radial oscillations. We show results obtained by applying this technique to a variety of surfaces, as well as experimental results for a spherical pendulum that shows reasonable agreement with theoretical predictions. The EAPEF is a useful tool for analyzing motion on surfaces of revolution, and can be used as the basis for interesting student projects.