Brachistochrone curve
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 level), but always starts at a cusp.
Earlier, in 1638, Galileo Galilei 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.
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".
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.
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., 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.
[13] 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.
In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital.
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.
[11] According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem.
[16] In a letter to Henri Basnage, held at the University of Basel Public Library, dated 30 March 1697, Johann Bernoulli stated that he had found two methods (always referred to as "direct" and "indirect") to show that the Brachistochrone was the "common cycloid", also called the "roulette".
This paper was largely ignored until 1904 when the depth of the method was first appreciated by Constantin Carathéodory, who stated that it shows that the cycloid is the only possible curve of quickest descent.
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.
He defines m so that MD = mx, and n so that Mm = nx + na and notes that x is the only variable and that m is finite and n is infinitely small.
For each point, M on the curve, the radius of curvature, MK is cut in 2 equal parts by its axis AL.
He has little time for our new analysis, describing it as false (He claims he has found 3 ways to prove that the curve is a cubic parabola)" – Letter from Johan Bernoulli to Pierre Varignon dated 27 Jul 1697.
According to Fermat’s principle, the actual path between two points taken by a beam of light (which obeys Snell's law of refraction) is one that takes the least time.
The equations above lead to two conclusions: Assuming for simplicity that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after falling a vertical distance D: Rearranging terms in the law of refraction and squaring gives: which can be solved for dx in terms of dy: Substituting from the expressions for v and vm above gives: which is the differential equation of an inverted cycloid generated by a circle of diameter D=2r, whose parametric equation is: where φ is a real parameter, corresponding to the angle through which the rolling circle has rotated.
In June 1696, Johann Bernoulli had used the pages of the Acta Eruditorum Lipsidae to pose a challenge to the international mathematical community: to find the form of the curve joining two fixed points so that a mass will slide down along it, under the influence of gravity alone, in the minimum amount of time.
At the suggestion of Leibniz, Bernoulli extended the challenge until Easter 1697, by means of a printed text called "Programma", published in Groningen, in the Netherlands.
This solution, later published anonymously in the Philosophical Transactions, is correct but does not indicate the method by which Newton arrived at his conclusion.
Bernoulli, writing to Henri Basnage in March 1697, indicated that even though its author, "by an excess of modesty", had not revealed his name, yet even from the scant details supplied it could be recognised as Newton's work, "as the lion by its claw" (in Latin, ex ungue Leonem).
The much quoted version tanquam ex ungue Leonem is due to David Brewster's 1855 book on the life and works of Newton.
[21] John Wallis, who was 80 years old at the time, had learned of the problem in September 1696 from Johann Bernoulli's youngest brother Hieronymus, and had spent three months attempting a solution before passing it in December to David Gregory, who also failed to solve it.
However, it is possible, with a high degree of confidence, to construct Newton's proof from Gregory's notes, by analogy with his method to determine the solid of minimum resistance (Principia, Book 2, Proposition 34, Scholium 2).
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.
Since the displacement EL is small, it differs little in direction from the tangent at E so that the angle EnL is close to a right-angle.
1 is the minimum curve not yet determined, with vertical axis CV, and the circle CHV removed, and Fig.
