Liouvillian Solutions in the Problem of Rolling of a Heavy Homogeneous Ball on a Surface of Revolution

The problem of a heavy homogeneous ball rolling without slipping on a surface of revolution is a classical problem of the nonholonomic system dynamics. Usually, when considering this problem, following the E.J. Routh approach, it is convenient to define explicitly the equation of the surface on which the ball’s center is moving. This surface is equidistant from the surface on which the contact point is moving. It is known from the classic works by Routh and F. Noether that, if a ball rolls on a surface such that its center moves along a surface of revolution, then the problem is reduced to solving the second-order linear differential equation. Therefore, it is of interest to study for which surfaces of revolution the corresponding second-order linear differential equation admits a general solution expressed by Liouvillian functions. To solve this problem, it is possible to apply the Kovacic algorithm to the corresponding second-order linear differential equation. In this paper, we present our own method to derive the corresponding second-order linear differential equation. In the case in which the center of the ball moves along an ellipsoid of revolution, we prove that the general solution of the equation is expressed through Liouvillian functions.