The Brachistochrone Curve

As with many technical terms in mathematics, the word brachistochrone originates from the Greek for ‘shortest time’. This curve has been rightly named, as you’ll see.

The brachistochrone curve was originally a mathematical problem posed by Swiss mathematician Johann Bernoulli in June 1696, and the problem is this: if a body is released from rest at a point A, what is the path of quickest descent for the body to take from point A to a lower point B under only the influence of gravity?

The Brachistochrone Problem.png

The shortest path is clearly a straight line from A to B, but this doesn’t necessarily correlate to the shortest time. A curved path, which starts steep and ends shallow, could provide a shorter time because the body reaches a higher maximum speed.

For mathematicians this is a very meaningful problem, because it’s easy to qualitatively describe; however finding a quantitative solution that works for all cases of the problem proves to be far more laborious. This is added onto the fact that, in maths, there are a countless number of ways to describe curves.

Bernoulli himself solved the problem in two weeks, but sadly he didn’t receive any responses to his problem within the six months he stipulated.

After extending the time by another year and a half at the request of the German mathematician Gottfried Leibniz, Bernoulli personally mailed the problem to Isaac Newton on 29th January of the following year. Although Newton was less than pleased to be challenged by someone he considered ‘beneath’ him, he stayed up until four in the morning, solving the problem impressively in one night. Newton mailed the solution back anonymously but Bernoulli recognised the author of the solution anyway, exclaiming that he ‘recognises a lion from its claw’.

In the end, four of the other greatest mathematicians of the generation responded with their respective solutions: Jakob Bernoulli (Johann’s brother, whom he incidentally had a petty rivalry with), Leibniz, l’Hôpital, and von Tschirnhaus.

Johann Bernoulli’s solution takes quite a creative approach, by using Fermat’s principle of least time. Fermat’s principle states that a ray of light will always take the path of shortest time (you can already anticipate the relation to the brachistochrone curve). Light travels more quickly in materials with a higher refractive index, and vice versa, so this principle has specific focus on when light crosses a material boundary.

A straight line may not be the quickest path, because the ray will want to spend less time in the material with a high refractive index and more time in the material with a low refractive index. For example, in the diagram below the bent path would be in fact quicker than the straight path, because light travels more slowly in water.

Fermat's Principle of Least Time.png

Bernoulli modelled his brachistochrone curve as a ray of light travelling through many, many layers of slightly varying refractive indices, that it essentially becomes a continuous spectrum of varying refractive indices. After some extensive calculus, the equations were bottled down to a differential equation, which Bernoulli supposedly instantly recognised as the differential equation of a cycloid. A cycloid is the path traced by a point on the circumference of a rolling circle:

cycloid
Image by Wikimedia Commons user D.328.

So as it turns out, the path of shortest time is not a straight line, nor the arc of a circle, nor a parabola, but a cycloid. The beauty of the brachistochrone curve lies in its generality; it works for any two points (even horizontal ones because we assume no resistive forces). And for closure, the ideal path is shown by the black curve in the diagram below:

the-brachistochrone-curve
Image by MAA.

The brachistochrone curve, due to the essence of the original problem, is a major consideration in many engineering designs. Winter sports, for instance skiing or skeleton, employ brachistochrone slopes to maximise chances of breaking world records. And in a world with an ever increasing need for speed, I’m sure you can think of plenty of other situations in which we want the path of quickest descent.

Yanhao

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