The unrestrained brachistochrone Available to Purchase

The unrestrained brachistochrone problem is to find the path of a frictionless track between two horizontally separated points along which a block with an initial velocity will travel in the shortest time. The track lies in a uniform gravitational field and must be horizontal at its beginning and its end. Furthermore, the block must slide along the track like a block on an inclined plane and it must remain in contact with the track at all times. Although the problem is well posed, the nature of the unusual constraints makes it quite difficult to solve using standard variational techniques, such as those of Euler and Lagrange. We present a solution that avoids explicit reliance on such techniques.