The Calculus Diaries - Jennifer Ouellette [82]
In 1696, Johann countered with a challenge to solve a particularly knotty conundrum: the problem of the brachistochrone. The word derives from the Greek words brachistos (“shortest”) and chronos (“time”). Johann cheated a little, having already solved the problem himself, but the challenge was deceptively simple on the surface. Assuming two fixed points, one higher than the other, what shape would a curved path between those points have to be for a rolling ball to reach the lower point the fastest? (In fine physics tradition, this problem assumes constant gravity and ignores friction.)
You might be tempted to dredge up a bit of long-forgotten knowledge from your high school geometry class and suggest that the shortest distance between two points is a straight line—ergo, a straight line is the fastest possible path. Resist that temptation. We are dealing with a curve in this instance. Galileo proposed in 1638 that the curve would be the arc of a circle; he, too, was mistaken. If we actually performed the experiment, it would quickly become clear that the steeper the curve between the two points, the faster the ball will gain speed.
Technically, this is a minimization problem: We are attempting to find the least possible amount of time it takes for the ball to descend. Yet because there is more than one quantity that is varying, the solution involves considering each and every possible path between the two points—truly a job for calculus. The solution is a cycloid, which is the curve created by a point on the rim of a wheel along a straight line.
Turn that path upside down—as with the inverted catenary curve that gives one the optimal shape for a stable arch—and you will get the path of fastest descent. You can test this result by building two tracks: one shaped like a cycloid, the other shaped like the arc of a circle, for comparison. Now roll two balls down each track simultaneously. The one on the cycloid path will reach the bottom first. Nor does it matter where one starts the ball along this curved path; it will still arrive at the bottom in precisely the same amount of time.
Five individuals solved the brachistochrone problem posed by Johann Bernoulli correctly: Johann himself; his brother Jakob; Guillaume François Antoine, Marquis de l’Hôpital; and the two founders of calculus, Leibniz and Newton. Newton was working as Master of the Mint at the time and received the challenge after a long day’s work. In general, Newton was loath “to be dunned [pestered] and teased by foreigners about mathematical things.” But the story goes that he was sufficiently intrigued by the problem that he stayed up until four A.M. until he solved it. All in all, it took him twelve hours. He submitted his solution anonymously to the Royal Society, but Johann was not fooled, claiming, “I recognize the lion by his print.”
In the process of uncovering the solution to this puzzle, the calculus of variations was born; Johann’s student, Leonhard Euler, refined his mentor’s techniques in 1766 and coined the term. This is calculus with an infinite number of variables. One would normally try to find the optimum value for a single variable (x), but in the case of the brachistochrone problem, it is necessary to integrate over all possible curves to find the optimal solution—selecting one curve from an infinite number of possibilities.
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