Section 1.5 The brachistochrone problem
¶Newton's problem, published in the Principia in 1686, lay apparently unnoticed for more than a decade before a new interest aroused by a second and more famous problem of the calculus of variations caused it to be studied again. It is not surprising that this happened because Newton's description of his results is very informal and concise. He gave no hint of a larger class of similar questions, and no suggestion of a method of solution which might have been applicable to such a class. To discover the beginnings of active research in the calculus of variations we must turn therefore to other writers. In the period which followed the discovery and publication of the calculus methods of Newton and Leibnitz two of the most prominent and successful researchers in the new analysis were the Swiss mathematicians James Bernoulli (1654-1705), professor of mathematics at the University of Basle, and his brother John (1667-1748). The younger brother was a student of the elder, and among those students he was in later years by far the most distinguished on account of his varied and successful researches. He studied with James until the year 1690 when he forsook Basle for travel and the study of mathematics in France. Shortly after his return he accepted in 1695 a professorship at the University of Groningen, and in 1705, upon the death of James Bernoulli, he returned to Basle to spend the remainder of his life as professor of mathematics in his native city. In the years just preceding 1695 a rivalry sprang up between the two brothers the reason for which is not well understood. It was at times amusingly undignified, and from the scientific standpoint unjustifiable, since both brothers were with somewhat different temperaments remarkably able and worthy of respect. Whatever may have been the cause of their dissension it is at any rate true that the friction between them gave an unusual impetus and zest to the beginnings of the calculus of variations. In June, 1696, John Bernoulli proposed his now famous brachistochrone problem, and publicly incited the mathematicians of the world to give it their consideration, according to a custom which was common at the time. We know that the problem aroused great interest and that Newton, Leibniz, and l'Hospital (1661-1704), besides the brothers Bernoulli, found the correct solution.
The problem of the brachistochrone (βραχιστος (brachistos) = shortest, χρονος (chronos) = time) is that of determining a path down which a particle will fall from one given point to another in the shortest time. Let the \(y\)-axis for convenience be taken vertically downward, as in Figure 1.5.1, the two fixed points being 1 and 2. The initial velocity \(v_1\) at the point 1 is supposed to be given. In Chapter III we shall see that for a curve defined by an equation of the form \(y=y(x)\) the time of descent from 1 to 2 is \(\frac{1}{\sqrt{2g}}\) times the value of the integral
where \(g\) is the gravitational constant and \(\alpha\) has the constant value \(\alpha = y_1-\frac{v_1^2}{2g}\text{.}\) The problem of the brachistochrone is then to find among the curves \(y=y(x)\) which pass through two points 1 and 2 one which minimizes the integral \(I\text{.}\)
The only discussions of the problem which were published in full in response to John Bernoulli's invitation were those of the Bernoulli brothers themselves, in May, 1697, and they are in many respects characteristic of their authors. John's paper is to this day most elegant and satisfactory reading. He saw that the curve of quickest descent is identical with the path of a ray of light in a medium with a suitably selected variable index of refraction, and a known property of such paths enabled him to attain very quickly and easily a solution. His method can be applied, however, to only a restricted class of similar questions. The solution of James was more laborious, and to us much less attractive, since it was couched in the language of the relatively clumsy geometrical analysis which preceded the invention of the calculus and which was commonly used for some time thereafter. But his method was a more general one than that of his younger brother and was the first step in a long series of researches which has led to the theory of the calculus of variations as we know it today.
At the close of his paper James invited mathematicians in general to consider a much more difficult problem of the calculus of variations which he had devised, and he offered to John in particular a money prize of fifty ducats for a satisfactory solution. As it turned out, however, the ducats were saved, for although John claimed to have done so he did not as a matter of fact succeed in his attacks upon the problem, and after a rather bitter discussion which dragged on for a number of years James finally published his own solution in 1701. The two papers of 1697 and 1701 of James Bernoulli were the starting-point for the researches of Euler (1707-83), a native of Basle and pupil of John Bernoulli, one of the greatest of the world's mathematicians. It is to Euler that we owe the first important result in the modern theory of the calculus of variations, as we shall see in later chapters.
It is fair to say that the theory of the calculus of variations had its beginning in the interesting brachistochrone problem of John Bernoulli. One should not infer from this remark that no problems of the calculus of variations were known earlier, for we have seen already that Newton had proposed such a problem and described a characteristic property of its solution. Furthermore the brachistochrone problem itself was more or less definitely in the mind of Galileo (1564-1642) in 1630 and 1638 when he compared the time of fall of a particle along an arc of a vertical circle with those along polygons inscribed in the arc. He seems to conclude that the time of descent on a circular arc is shorter than the times on all other paths joining its endpoints, but his proof does not justify this result. Nowadays we know that the solution curve is neither a circle nor a straight line but a cycloid, as will be proved in Chapter III. A still older problem of the calculus of variations is the isoperimetric problem of the ancient Greeks described in Section 1.7 below. None of these, however, could rightly be regarded as the starting-point of the theory of the calculus of variations, for in the early references to them there were no indications of other problems of similar type, or of methods of solution possessing generality of application.