Existence of Liouvillian Solutions in the Problem of Rolling Motion of a Dynamically Symmetric Ball on a Perfectly Rough Sphere

Resumo

The problem of rolling without sliding of a rotationally symmetric rigid body on a motionless sphere is considered. The rolling body is assumed to be subjected to forces whose resultant force is applied to the center of mass G of the body, directed to center O of the sphere, and depends only on the distance between G and O. In this case, the process of solving this problem is reduced to integrating the second-order linear differential equation with respect to the projection of the angular velocity of the body onto its axis of dynamic 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 body 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 to this equation for the case when the rolling rigid body is a dynamically symmetric ball.