The Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint

This paper investigates the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that the sphere can slip in the direction of the projection of the symmetry axis onto the supporting plane. Equations of motion are obtained and their first integrals are found. It is shown that in the general case the system considered is nonintegrable and does not admit an invariant measure with smooth positive density. Some particular cases of the existence of an additional integral of motion are found and analyzed. In addition, the limiting case in which the system is integrable by the Euler – Jacobi theorem is established.