The ellipsoidal pendulum

The problem of the motion of a heavy particle on a weightless inextensible thread, attached at two points on a horizontal level with a slack is considered. The configuration space of this system is a spheroid. The equations of the trajectories of a heavy particle on the surface of the spheroid with a horizontal axis of revolution with a specified energy level are obtained in dimensionless form. A solution of the equations is given in quadratures for a fairly small value of the ratio of the maximum possible potential energy of the particle on the spheroid to its total energy. The conditions for the motion of the particle to be stable along the spheroid equator and the conditions for the particle release from the constraint are determined.