On parametric representation of brachistochrone problem with Coulomb friction

The motion of a material point along a trajectory located in a vertical plane and connecting two given fixed points is considered. It is assumed that the motion occurs in a homogeneous field of gravity and taking into account the Coulomb friction caused by statical and dynamical reactions of the support curve. The classical problem of finding a curve of fastest descent for a material point sliding with friction from one fixed point to another, that is, a trajectory for which the descent time is the shortest, i.e. the problem of brachistochrone, is discussed. A parametric method for solving the brachistochrone problem with Coulomb friction is proposed and the solution of the variational problem is given in the form of quadratures.