Existence of liouvillian solutions in the problem of motion of a dynamically symmetric ball on a perfectly rough sphere

Abstract

The problem of rolling without sliding of a rotationally symmetric rigid body on a sphere is considered. The rolling body is assumed to be subjected to the forces, the resultant of which is directed from the center of mass G of the body to the center O of the sphere, and depends only on the distance between G and O. In this case, the solution of this problem is reduced to solving the second-order linear differential equation over the projection of the angular velocity of the body onto its axis of symmetry. Using the Kovacic algorithm, we search for Liouvillian solutions of the corresponding second order linear differential equation. We prove that all solutions of this equation are Liouvillian in the case when the rolling rigid body is a nonhomogeneous dynamically symmetric ball. The paper is organized as follows. In the first paragraph, we briefly discuss the statement of the general problem of motion of a rotationally symmetric rigid on a perfectly rough sphere. We prove that this problem is reduced to solving the second order linear differential equation. In the second paragraph, we find Liouvillian solutions of this equation for the case, when the rolling rigid body is a dynamically symmetric ball.