Liouvillian solutions in the problem of rolling of a heavy homogeneous ball on a surface of revolution

The problem of rolling without sliding of a homogeneous ball on a fixed surface under the action of gravity is a classical problem of 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 centre is moving. This surface is equidistant to the surface, over which the contact point is moving. From the classical works of E. J. Routh and F. Noether it was known that if the ball rolls on a surface such that its centre moves along a surface of revolution, then the problem is reduced to solving the second order linear differential equation. Therefore it is interesting to study for which surface of revolution the corresponding second order linear differential equation admits Liouvillian solutions. 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 when the centre of the ball moves along the ellipsoid of revolution we prove that the corresponding second order linear differential equation admits a liouvillian solution.