brachistochrone problem


The brachistochrone problem is a problem with which Johann Bernouilli challenged his contemporaries in Acta Eruditorum in June 1696:


Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise... Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.


Isaac Newton reportedly solved the problem between four in the evening and four in the morning after a hard day at the Royal Mint, later commenting: "I do not love to be dunned [pestered] and teased by foreigners about mathematical things..." Other correct solutions came in from Gottfried Leibniz, the Frenchman Guillaume de L'H˘pital, and Johann's brother Jakob. They, like Johann, realized that the solution to the brachistochrone problem, as it was also to the tautochrone problem, was a curve known as the cycloid. See also calculus of variations.