Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction Eric Weisstein's World of Physics) from one point to another in the least time. The brachistochrone problem was one of the earliest problems posed in the calculus of variations. Eric Weisstein's World of Math The solution, a segment of a cycloid, Eric Weisstein's World of Math was found by Leibniz, Eric Weisstein's World of Biography L'Hospital, Eric Weisstein's World of Biography Newton, Eric Weisstein's World of Biography and the two Bernoullis. Johann Bernoulli Eric Weisstein's World of Biography solved the problem using the analogous one of considering the path of light refracted by transparent layers of varying density. Note that bead may actually travel uphill along the cycloid for a distance, but the path is nonetheless faster than a straight line or any other line.
