He introduced the problem as follows:- I, Johann Bernoulli , address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal , Fermat , etc. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise. The problem he posed was the following:- 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. Johann Bernoulli was not the first to consider the brachistochrone problem.

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History[ edit ] Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc.

If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise Bernoulli wrote the problem statement as: 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. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid. Bernoulli allowed six months for the solutions but none were received during this period.

At the request of Leibniz, the time was publicly extended for a year and a half. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he "recognizes a lion from his claw mark". In his paper, Jakob Bernoulli gave a proof of the condition for least time similar to that below before showing that its solution is a cycloid. In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called in the calculus of variations.

Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus. Earlier, in , Galileo had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his Two New Sciences. He draws the conclusion that the arc of a circle is faster than any number of its chords, [13] From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle.

Consequently the nearer the inscribed polygon approaches a circle the shorter is the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the same.

Just after Theorem 6 of Two New Sciences, Galileo warns of possible fallacies and the need for a "higher science". In this dialogue Galileo reviews his own work. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics. Following advice from Leibniz, he only included the indirect method in the Acta Eruditorum Lipsidae of May He writes that this is partly because he believed it was sufficient to convince anyone who doubted the conclusion, partly because it also resolved two famous problems in optics which "the late Mr.

Huygens" had raised in his treatise on light. In the same letter he criticises Newton for concealing his method. In addition to his indirect method he also published the five other replies to the problem that he received. The method is to determine the curvature of the curve at each point. It was only in that Bernoulli explained how he solved the brachistochrone problem by his direct method.

According to him, the other solutions simply implied that the time of descent is stationary for the cycloid, but not necessarily the minimum possible. Analytic solution[ edit ] A body is regarded as sliding along any small circular arc Ce between the radii KC and Ke, with centre K fixed.

The first stage of the proof involves finding the particular circular arc, Mm which the body traverses in the minimum time. Of all the possible circular arcs Ce, it is required to find the arc Mm which requires the minimum time to slide between the 2 radii, KM and Km. To find Mm Bernoulli argues as follows. The small time to travel along arc Mm is M.

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## The brachistochrone problem

History[ edit ] Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise Bernoulli wrote the problem statement as: 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. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid. Bernoulli allowed six months for the solutions but none were received during this period. At the request of Leibniz, the time was publicly extended for a year and a half.

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## Brachistochrone Problem

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