On a toroidal pendulum

Abstract

We consider the problem of motion of a heavy particle on the surface of a torus with horizontal axis of rotation. On nondevelopable surfaces other than surfaces of revolution with vertical axis, the solution is known only for the surface of an elliptic paraboloid [1]. To solve the problem on the surface of a torus with horizontal axis of rotation, we use the method of reduction of equations of motion proposed in [2]. We construct the asymptotics of the general and periodic solutions and show that one can use this asymptotics when studying the motion of a heavy particle on an elliptic torus. We obtain the stability conditions in the first approximation for the particle motion along the outer equator and the lower meridian of the torus.