The brachistochronic motion of a heavy ball rolling along an imperfect rough surface

Abstract

The problem of brachistochronic motion of a heavy uniform ball rolling without slip along the upper outside surface of an imperfect rough stationary sphere, is solved. The control forces are located in the tangential plane, and their total power equals zero. In the first part of the paper the determination of the brachistochronic motion is solved as the problem of optimal control using Pontryagin’s maximum principle. This solution corresponds to the motion of the heavy ball along a perfect rough sphere. The second part provides the case when the constraint between the sphere and the ball is imperfectly rough. Here, the problem of optimal control is formulated in such way that the tangential component of the reaction of constraint is taken for the control, with the restriction resulting from Coulomb’s laws of sliding friction. The problem thus formulated belongs to the theory of singular optimal controls, and the solution that satisfies the Maximum principle consists of a singular part and a nonsingular part.