Motion of chaplygin ball on an inclined plane

Abstract

The rolling motion of a dynamically nonsymmetricbalanced ball (Chaplygin ball) on an inclined plane isstudied. For the case of a horizontal plane, Chaplygindemonstrated this problem to be integrable. For a non-zero slope, the system is integrable only if the motionstarts from a state of rest (E.N. Kharlamova). It isshown that, in the general case, the system exhibits arather simple asymptotic behavior.Kharlamova [1] found an integrable case for theequations of motion of a dynamically nonsymmetricbalanced ball on an inclined plane in a gravitationalfield when the motion starts from a state of rest. Similarto the classical work of Chaplygin [2], in paper [1], it isassumed that no sliding occurs at the contact point andthe problem is described by the equations of nonholo-nomic mechanics. For the case of a zero slope, Chaply-gin found the first integrals and the invariant measure,which are necessary for the intregrability (according tolast multiplier theory and to Euler–Jacobi), he also inte-grated the equations of motion using sphero-conicalcoordinates. In Kharlamova’s case, the equations ofmotion of a ball on an inclined plane can be reduced tothe conventional equations of motion of a Chaplyginball (without slope) by a suitable change of the coordi-nates and time. We note that, in [1], the presentation israther sophisticated; for this reason, we present our ownderivation of the results of [1].Let a heavy dynamically nonsymmetric balancedball of radius R roll without sliding on an inclinedplane. Choose a fixed coordinate system Oxyz with thez-axis perpendicular to the inclined plane and the x- andy-axes determined by the intersection of the inclinedplane with the vertical and horizontal planes, respec-tively (Fig. 1). We fix a moving coordinate system withthe principal axes of the ball so that the inertia tensor isI = diag(I