brachistochrone curve

This solution was later used to attack the problem of the brachistochrone curve.

Hence, the brachistochrone curve is tangent to the vertical at the origin.

The tautochrone curve is the same as the brachistochrone curve for any given starting point.

The calculus of variations may be said to begin with the brachistochrone curve problem raised by Johann Bernoulli (1696).

Jakob Bernoulli and Johann Bernoulli solve the brachistochrone curve problem, the first result in the calculus of variations.

He first got a taste for maths by reading the Géométrie de l'infini of Fontenelle and gave solutions to the problems of the tautochrone curve, the brachistochrone curve and orthogonal trajectories.

In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).

Given two points A and B, with A not lower than B, only one upside down cycloid passes through both points, has a vertical tangent line at A, and has no maximum points between A and B: the brachistochrone curve.

In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the brachistochrone curve, the curve that minimizes the time it takes for a falling particle to travel from its start to its end.