Investigation of the Stability of Stationary Motions in a System of Two Rolling Spheres

This study investigates the dynamics of a nonholonomic mechanical system consisting of two rigid spherical bodies. The primary configuration involves the base sphere rolling without slipping on the horizontal plane which is fixed or moves uniformly and in a straight line. Another sphere rolls, also without slipping, along the external or internal surface of the base sphere. A system of equations of motion is constructed. A complete set of first integrals and an invariant measure are identified. It is demonstrated that the system of equations of motion is integrable by virtue of the Euler – Jacobi theorem and is therefore reducible to quadratures. To analyze the stability of the system’s stationary motions, a reduced potential energy function is derived. It is shown that, when the sphere moves along the internal surface of the base sphere, the stationary motions are orbitally stable with respect to perturbations that preserve the values of the first integrals. The conditions for orbital stability are established for motion along the external surface of the base sphere. To prove stability with respect to arbitrary perturbations, other methods are required.