Brachistochrone with Coulomb friction

The classical brachistochrone is considered with the inclusion of a resistant force, which is due to Coulomb friction, in addition to the uniform gravitational force that is present. The solution to this problem is expressed in terms of standard functions, and it is developed in two separate ways by means of constrained variational calculus methods. These ways involve formulations of the problem in terms of temporal and spatial independent variables, respectively. The equations of motion that result in both cases are non-linear and coupled. The utilization of path variables is a central feature of the developments provided.