The general unrestrained brachistochrone Available to Purchase

The general unrestrained brachistochrone problem is to find the frictionless track between two points in a uniform gravitational field along which a particle with initial velocity will slide in the shortest time. Particularly important is the condition that the particle remain in contact with the track, even though it is unrestrained to the track. That is, the particle must slide along the track like a block on an inclined plane, not like a bead on a wire. Because of the unusual nature of the constraints, the techniques of Euler and Lagrange cannot be applied to this problem as it stands; a solution is presented here that does not rely on such an approach. The conditions imposed by the initial and final positions and velocities fall into our four classes, each having a unique form of solution, consisting of sections of free‐fall parabolas and cycloids.