Brachistochrone problem. The classical problem in calculus of variation is the so called brachistochrone problem1 posed (and solved) by Bernoulli in The brachistochrone problem asks us to find the “curve of quickest descent,” and so it would be particularly fitting to have the quickest possible solution. THE BRACHISTOCHRONE PROBLEM. Imagine a metal bead with a wire threaded through a hole in it, so that the bead can slide with no friction along the .

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If kinetic friction is included, the problem can also be solved analytically, although the solution is significantly messier. This story gives some idea of Newton’s power, since Johann Bernoulli took brachixtochrone weeks to solve it.

The first stage of the proof involves finding the particular circular arc, Mm which the body traverses in the minimum time. In a letter to Henri Basnage, held at the University of Basel Public Library, dated 30 MarchJohann Bernoulli stated that he had found 2 methods always referred to as “direct” and “indirect” to show that the Brachistochrone was the “common cycloid”, also called the “roulette”.

In the same letter he criticises Newton for concealing his method. Hints help you try the next step on your own.

Since it appears that the body is moving upwards from e to E, it must be assumed that a small body is released from Z and slides along the curve to A, without friction, under the action of gravity. The time to travel from a point to another point is given by the integral. Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus. If the arc, Cc subtended by the angle infinitesimal angle MKm on IJ is not circular, it must be greater than Ce, since Cec becomes a right-triangle in the limit as angle MKm approaches zero.


In addition to his indirect method he also published the five other replies to the problem that he received. History of Mathematics, Vol. From this the equation of the curve brachistocgrone be obtained from the integral calculus, though he does not demonstrate this.

Newton was challenged to solve the problem inand did brachistochrine the very next day Boyer and Merzbachp. The brachistochrone problem was one of the earliest problems posed in the calculus of variations.

Brachistochrone problem

This page was last edited on 23 Decemberat Assume that it traverses the straight line eL to point L, horizontally displaced from E by a small distance, o, instead of the arc eE. Isaac Newton January “De ratione temporis quo grave labitur per rectam data duo puncta conjungentem, ad tempus brevissimum quo, vi gravitatis, transit ab horum uno ad alterum per arcum cycloidis” On a proof [that] the time in which a weight slides by a line joining two given points [is] the shortest in terms of time when it passes, via gravitational force, from one of these [points] to the other through a cycloidal arcPhilosophical Transactions of the Royal Society of London 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.

It seems highly suspicious that it would take so long for a communication from Groningen to arrive in London. Bernoullio, deinde a Dn. Therefore, the increase in time to traverse a small arc displaced at one endpoint depends only on the displacement at the endpoint and is independent of the position of the arc.

The brachistochrone problem

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. The speed at any point is given by a simple application of conservation of energy equating kinetic energy to gravitational potential energy.


When Jakob correctly did so, Johann tried to substitute the proof for his own Boyer and Merzbachp. Johann Bernoulli’s direct method is historically important as it was the first proof that the brachistochrone is the cycloid. In this dialogue Galileo reviews his own work. Newton, claimed he had been unaware of the challenge until he first saw it at 4 pm on 29 January, some five weeks after its publication. A History of Mathematics, 2nd ed.

Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics. This was 22 December in the Julian Calendar, in use in Britain. In fact, the solution, which is a segment of a cycloidwas found by Leibniz, L’Hospital, Newton, and poblem two Bernoullis. Bernoulli allowed six months for the solutions but none were received during this period.

Assume AMmB is the part of the cycloid joining A to B, which the body slides down in the minimum time. Because eEFf is the minimum curve, t — T is must be greater than zero, whether o is positive or negative.

Quick! Find a Solution to the Brachistochrone Problem

More specifically, the brachistochrone can use up to a complete rotation of the cycloid at the limit when A and B are at the same levelbut always starts at a cusp. University of Chicago Press, pp.

In the solution, the bead may actually travel uphill along the cycloid for a distance, but the path is nonetheless faster than a straight line or any other line.

However, the function is particularly nice since does not appear explicitly.