In mathmatics and physics, a brachistochrone curve meaning "shortest time" or curve of fastest descent, is the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. Incidentally, for a given starting point, the brachistochrone curve is the same as the tautochrone curve. More specifically, the solution to the brachistochrone and tautochrone problem are one and the same, the cycloid.
The problem can be solved with the tools from the calculus of variations and optimal control
The curve is independent of both the mass of the test body and the (local) strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B. If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time will differ from the one described above. The caculation part is below
The problem can be solved with the tools from the calculus of variations and optimal control
The curve is independent of both the mass of the test body and the (local) strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B. If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time will differ from the one described above. The caculation part is below