Today I want to share with you one very fun application of optimization. Formally, it is called the Brachistochrone Problem, yet people can avoid remembering this long name by remembering the problem as “How to fall the fastest”. The problem was undoubtedly one of the most interesting at the time of introduction, and I thought it would be interesting to know the history behind one of probably the simplest applications of mathematics of all time.
- History of the problem:
For anyone who is interested, the name “Brachistochrone” is actually an Ancient Greek term meaning “shortest time”. The original statement of the problem, as posted by Johann Bernoulli in the first scientific journal of Europe Acta Eruditorum in June 1696, was:
“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.”
Funnily enough, the problem was enclosed in a rather provocative challenge statement:
“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. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.”
Of course, such an obvious natural phenomenon such as falling could not have been neglected all the way until 1696. Before that, Galileo Galilei, the same person who had proven that the falling speed for all objects is the same if not accounting for the air drag, had also considered the problem in 1638. However, due to calculus not having developed back then, Galileo’s solution of a circle arc passing through A and B was close, yet incorrect.
Image 1: Illustrations of various considered curves (Source: maa.org)
In the end, 5 mathematicians responded to Johann’s problem statement: Isaac Newton, Jakob Bernoulli (Johann’s brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l’Hôpital. One of them in particular, Isaac Newton, showed his prowess by solving the problem in just a single day, and not only was he incorrect, his solution was also so unique that Johann recognized the author from the first sight. Regardless, the great number of responses to the problem did prove the extent of the development of calculus just 60 years after Galileo’s era, and paced the path for mathematics, especially optimization research, for years to come.
- Solution to the problem:
The answer for the problem is what would be later called a Brachistochrone curve, a version of the cycloid with the cusps pointing upward, starting at A – the beginning point – and ending at B – the destination point. In turn, the cycloid is a curve equivalent to the path of a point on a circle when it is rolling on a surface:
Image 2: How a cycloid is formed
There are various approaches to the problem, but almost all of them consider the velocity as the derivative of distance with regard to time. This particular consideration was undoubtedly one of the very first applications of calculus of variations (in this case, the distance) in order to find extremes (in this case, the smallest amount of time needed to reach the destination). It was Jacob Bernoulli’s solution and expansion of the problem that Leonhard Euler developed calculus of variations from, and later Joseph-Louis Lagrange, a famous French mathematician, would step in to create the modern infinitesimal calculus. The importance of the Brachistochrone problem, therefore, is to be emphasized and recognized throughout the history of Mathematics.
Crash on Maths 