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.

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.

We have generalized the unrestained brachistochrone problem to include rolling objects in order to construct a demonstration brachistochrone track. Due to the nature of the constraints this well-posed extremal problem cannot be solved using the techniques of Euler and Lagrange. The solution, parametrized by a measure of the generalized cylinder’s moment of inertia, reduces to that for the (sliding) unrestrained brachistochrone in the limit of vanishing moment of inertia.