Brachistochronic motion of a multibody system with Coulomb friction

Brachistochrone motion of a mechanical system subject to the action of constraints with Coulomb friction in the homogeneous field of gravity forces is considered. In case that it is possible to determine the forces of Coulomb friction as functions of generalized coordinates, velocities and accelerations, the constrained variation problem which solution represents the motion asked is formulated. Especially, when the functional does not depend on generalized coordinates, characteristics of differential equations of brachistochrone motion are analyzed and it is found that in case of an even number of degrees of freedom of motion they have at least one solution with the form of the first integral, as their algebraic consequence. This shows the complete analogy between the brachistochrone of a system with two degrees of freedom and the brachistochrone of a particle with Coulomb friction. Also, it is shown that the famous Euler's extension of the Bernoulli's brachistochrone to the case of motion of a particle in the homogeneous field of gravity forces and the field of forces of viscous friction represents a special case of results of this paper. The paper is illustrated with an example of a system of rigid bodies with two degrees of freedom of motion.