Brachistochrone(redirected from Brachistochrone curve)
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the curve of most rapid descent—that is, the one of all possible curves connecting two given points A and B of a potential force field that a mass point moving along the curve with an initial velocity equal to zero and acted upon only by the forces of the field will traverse from position A to position B in the shortest time. When the movement occurs in a homogeneous gravitational field, the brachistochrone is a cycloid with a horizontal base and a point of return that coincides with point A. The solution of the brachistochrone problem (Johann Bernoulli, 1696) served as the starting point for the development of the calculus of variations. The error of Galileo, who tried to prove that the brachistochrone is a circumferential arc, is instructive. (See G. Galilei, Izbrannye trudy, vol. 2, Moscow, 1964, pp. 298-301, note 465.)