Contribution to the brachistochrone problem with Coulomb friction

This paper formulates and solves in closed form (expressed by elementary functions) the brachistochrone problem with Coulomb friction of a particle which moves down a rough curve in a uniform gravitational field assuming that the initial velocity of the particle is different from zero. The problem is solved by the application of variational calculus. Two variants are considered: first, the initial position and the final position of the particle are given; second, the initial position is given, and the final position lies on a given vertical straight line. The new approach in treating this problem by variational calculus lies in the fact that the projection sign of the normal reaction force of the rough curve onto the normal to the curve is introduced as the additional constraint in the form of an inequality. This inequality is transformed into an equality by introducing a new state variable. Although this is fundamentally a constrained variational problem, by further introducing a new functional with an expanded set of unknown functions, it is transformed into an unconstrained problem where broken extremals appear. Brachistochrone equations in parametric form are obtained for both variants which are examined, with the slope angle of the tangent to the brachistochrone being taken as the parameter. These equations contain a certain number of unknown constants which are determined from the corresponding systems of nonlinear algebraic equations. They are solved by an alternative approach which is based on the application of differential evolution. The obtained brachistochrones are generally two-segment curves with the initial line segment representing a free-fall parabola in nonresistant medium. It is shown that regarding the special values of the parameters the results of the paper coincide with the known results from literature.