In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead from rest slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696 and famously solved in one night by Isaac Newton in 1697, though Bernoulli and several others had already found solutions of their own months earlier.
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. But the portion of the cycloid used for each varies. Specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp, while the tautochrone can use only up to the first half rotation and always ends at the horizontal.Stewart, James. "Section 10.1 - Curves Defined by Parametric Equations." Calculus: Early Transcendentals. 7th ed. Belmont, CA: Thomson Brooks/Cole, 2012. 640. Print. The problem can be solved using tools from the calculus of variations and optimal control.Ross, I. M. The Brachistochrone Paradigm, in Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. .
The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B.Hand, Louis N., and Janet D. Finch. "Chapter 2: Variational Calculus and Its Application to Mechanics." Analytical Mechanics. Cambridge: Cambridge UP, 1998. 45, 70. Print. If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.
History
Galileo's problem In 1638, Galileo Galilei 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, This conclusion had appeared six years earlier in Galileo's Dialogue Concerning the Two Chief World Systems (Day 4).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.
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Consequently the nearer the inscribed polygon approaches a circle the shorter 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, he 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.
Galileo's conjecture is that "The shortest time of all [for a movable body] will be that of its fall along the arc ADB [of a quarter circle] and similar properties are to be understood as holding for all lesser arcs taken upward from the lowest limit B."
In Fig.1, from the "Dialogue Concerning the Two Chief World Systems", Galileo claims that the body sliding along the circular arc of a quarter circle, from A to B will reach B in less time than if it took any other path from A to B. Similarly, in Fig. 2, from any point D on the arc AB, he claims that the time along the lesser arc DB will be less than for any other path from D to B. In fact, the quickest path from A to B or from D to B, the brachistochrone, is a cycloidal arc, which is shown in Fig. 3 for the path from A to B, and Fig.4 for the path from D to B, superposed on the respective circular arc.
Introduction of the problem Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696.Johann Bernoulli (June 1696) "Problema novum ad cujus solutionem Mathematici invitantur." (A new problem to whose solution mathematicians are invited.), Acta Eruditorum, 18 : 269. From p. 269: "Datis in plano verticali duobus punctis A & B (vid Fig. 5) assignare Mobili M, viam AMB, per quam gravitate sua descendens & moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B." (Given in a vertical plane two points A and B (see Figure 5), assign to the moving [body] M, the path AMB, by means of which — descending by its own weight and beginning to be moved [by gravity] from point A — it would arrive at the other point B in the shortest time.)Solutions to Johann Bernoulli's problem of 1696: Isaac Newton (January 1697) "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 arc), Philosophical Transactions of the Royal Society of London, 19 : 424-425. G.G.L. (Gottfried Wilhelm Leibniz) (May 1697) "Communicatio suae pariter, duarumque alienarum ad edendum sibi primum a Dn. Jo. Bernoullio, deinde a Dn. Marchione Hospitalio communicatarum solutionum problematis curva celerrimi descensus a Dn. Jo. Bernoullio Geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea propositi." (His communication together with [those] of two others in a report to him first from Johann Bernoulli, [and] then from the Marquis de l'Hôpital, of reported solutions of the problem of the curve of quickest descent, [which was] publicly proposed by Johann Bernoulli, geometer — one with a solution of his other problem proposed afterward by the same [person].), Acta Eruditorum, 19 : 201–205. Johann Bernoulli (May 1697) "Curvatura radii in diaphanis non uniformibus, Solutioque Problematis a se in Actis 1696, p. 269, propositi, de invenienda Linea Brachystochrona, id est, in qua grave a dato puncto ad datum punctum brevissimo tempore decurrit, & de curva Synchrona seu radiorum unda construenda." (The curvature of [light] rays in non-uniform media, and a solution of the problem [which was] proposed by me in the Acta Eruditorum of 1696, p. 269, from which is to be found the brachistochrone line [i.e., curve], that is, in which a weight descends from a given point to a given point in the shortest time, and on constructing the tautochrone or the wave of [light] rays.), Acta Eruditorum, 19 : 206–211. Jacob Bernoulli (May 1697) "Solutio problematum fraternorum, … " (A solution of [my] brother's problems, … ), Acta Eruditorum, 19 : 211–214. Marquis de l'Hôpital (May 1697) "Domini Marchionis Hospitalii solutio problematis de linea celerrimi descensus" (Lord Marquis de l'Hôpital's solution of the problem of the line of fastest descent), Acta Eruditorum, 19 : 217-220. reprinted: Isaac Newton (May 1697) "Excerpta ex Transactionibus Philos. Anglic. M. Jan. 1697." (Excerpt from the English Philosophical Transactions of the month of January in 1697), Acta Eruditorum, 19 : 223–224. He said:
Bernoulli wrote the problem statement as:
Johann and his brother Jakob Bernoulli derived the same solution, but Johann's derivation was incorrect, and he tried to pass off Jakob's solution as his own. Johann published the solution in the journal in May 1697, and noted that the solution is the same curve as Huygens' tautochrone curve. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid, but his proof is marred by his use of a single constant instead of the three constants, vm, 2g and D, below.
Bernoulli allowed six months for solutions but none were received during that period. At Leibniz's request, the time was publicly extended for a year and a half. At 4 p.m. on 29 January 1697, when he arrived home from the Royal Mint, Isaac Newton found the challenge in a letter from Johann Bernoulli. Newton stayed up all night to solve it and mailed the solution anonymously by the next post. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he "recognizes a lion from his claw mark". This gives some idea of the extent of Newton's exceptionally broad knowledge, intellect, and ability, since it took Bernoulli himself two weeks to solve it,D.T. Whiteside, Newton the Mathematician, in Bechler, Contemporary Newtonian Research, p. 122. and many of Europe's greatest mathematicians took months to solve it. Newton also wrote, "I do not love to be dunned and teased by foreigners about mathematical things", and had already solved Newton's minimal resistance problem, which is considered the first of its kind in the calculus of variations. Bernoulli had employed the principle of least time in deriving his solution, but not the calculus of variationsNewton had employed both, and wound up pioneering the field as a result of his work on the two problems.
In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l'Hôpital. Four of the solutions (excluding L'Hôpital's) were published in the same edition of the journal as Bernoulli's. 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. According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that Leonhard Euler refined into what Euler called (in 1766) the calculus of variations. Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus.
Johann Bernoulli's solution
Introduction
In a letter to L'Hôpital, (21/12/1696), Bernoulli wrote that in considering the problem of the curve of quickest descent, he first noticed a curious affinity or connection with another no less remarkable problem, leading to an "indirect method" of solution, and that shortly afterward he discovered a "direct method".
Direct method
In a letter to Henri Basnage, held at the University of Basel Public Library, dated 30 March 1697, Bernoulli wrote that he had found two methods (always called "direct" and "indirect") to show that the Brachistochrone was the "common cycloid", also called the "roulette". On Leibniz's advice, he included only the indirect method in the Acta Eruditorum Lipsidae of May 1697. He wrote that this was partly because he believed it was sufficient to convince anyone who doubted the conclusion and partly because it also resolved two famous problems in optics that "the late Mr. Huygens" had raised in his treatise on light. In the same letter he criticised Newton for concealing his method.
In addition to his indirect method, he published the five other replies to the problem he had received.
Bernoulli's direct method is historically important as a proof that the brachistochrone is the cycloid. The method is to determine the curvature of the curve at each point. All the other proofs, including Newton's (which was not revealed at the time), are based on finding the gradient at each point.