Brachistochronic motion of a nonholonomic rheonomic mechanical system

Abstract

The brachistochrone problem of the rheonomic mechanical system whose motion is subject to nonholonomic constraints is solved with nonlinear differential equations of motion. Apart from control forces, the system is influenced by the action of other known potential and nonpotential forces as well. The problem of optimal control is solved by applying Pontryagin's Maximum Principle and the singular optimal control theory. This procedure results in the two-point boundary value problem for the system of ordinary nonlinear differential equations of the first order, with a corresponding number of initial and end conditions. This paper determines the control forces that are realized by imposing on the system a corresponding number of independent ideal holonomic constraints, without the action of active control forces. These constraints must be in accordance with the previously determined brachistochronic motion. The method is illustrated with a single complex example that represents the first known concrete demonstration of brachistochronic motion of the nonholonomic rheonomic mechanical system.