The Mystery of the Fastest Slide: Can You Crack the Brachistochrone Code?

Unraveling the puzzle that sparked a different approach to understanding the dynamics of a physical system.

“If I have seen further, it is by standing on the shoulders of giants.” - Isaac Newton

Greetings, fellow Bohron

“Have you ever taken a thrilling ride on a roller coaster or experienced exhilaration at a skating rink? Or have you tried downhill skiing to quench your thirst for adventure?”

If so, brace yourself for a voyage back more than 300 years, which will reveal the exciting physics behind the euphoria you feel as you cover distances in the least amount of time.

Roller coasters are an excellent application of the Brachistochrone curve. Source

Brachistochrone - a seemingly complex word

Back in June 1696, Johann Bernoulli, one of the pioneers in the field of calculus, posed one of the most famous problems to the scientific community of the time, The Brachistochrone Problem. The problem statement is as follows:

Given two vertically separate points in a plane, what is the curve traced by a particle acted only by gravity that starts from rest at one of the points and travels to the other in the shortest time?

To simplify this seemingly complex word, let’s dive into the etymology of the word 'brachistochrone.' 'Brachistochrone’ originates from Greek—'brakhistos', meaning 'shortest', combined with 'khronos', meaning 'time.' Altogether, it refers to a curve of the shortest time or in other words, a curve of the fastest descent.

Now, let us brainstorm together to solve the problem posed by Bernoulli, assuming that you have never taken a course on calculus.

Straight line - a seemingly correct answer

Let me ask you a question: If a bead is to roll down a stiff wire connecting two vertically separate points and reach the finish in the briefest time possible, what shape should the wire have?

Different Path Options for a bead to travel between the points. Source

• The first curve shows a straight line. It is the shortest distance between two points, and it seems reasonable to assume that this curve would be the quickest route. 

• The second curve shows a parabolic (just a fancy word) path. The curve looks like a natural depiction of the bead falling under gravity.

• The third curve has an advantage as the bead can pick speed during the initial drop, and the bead transitions into the horizontal section with this increased speed.

• Or, do you think it is unreasonable to assume that path decides the time taken by the bead to travel down the wire? Is it that the time taken by bead along all paths is the same?

To elude this tricky path of trial and error, it's time to start our voyage from the initial point. Approximately 28 years before Bernoulli posed the question, Galileo Galilei, a great polymath, tried his hands on the Brachistochrone problem and arrived at an answer. Although the answer was incorrect, we can certainly take help from his work.

Galileo started by fixing point A in the plane and attempted to find another point B on the vertical line. In this configuration, AB represents the straight-line path a point mass could cover in the shortest time. He correctly calculated that this line would lie at an angle of 45 degrees with the vertical line.

Source

Galileo also demonstrated that the time taken by a point mass to travel along ABG is less than the time it takes to travel along AG. Similarly, the path ABCG is quicker than ABG, and ABCDG is quicker than ABCG, etc. In this way, he showed that the path along the circular arc is faster than any set of chords.

Source

We can trust Galileo’s calculations that show that a straight line is not the quickest path. So, which path is the quickest then?

The final piece of the puzzle

When Bernoulli publically posed the problem, he allowed six months for solutions. With none received, at Gottfried Wilhelm Leibniz's request, who was also a great polymath, Bernoulli extended the period to a year and a half. By the end of the time period, five mathematicians came up with solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l'Hôpital. 

Bernoulli found two solutions to this problem: a direct solution and an indirect solution. The direct method is analytical, whereas the indirect method is intuitive. We are going to discuss the indirect method. Instead of considering a particle subject to gravity, Bernoulli started with consideration of a light ray traveling through different media. Why?

Read on, and you will find the answer.

Bernoulli’s solution to the famous problem

The key to finding the solution to the problem is Fermat’s principle.

Fermat’s principle states that as light travels across a boundary between two media, it takes the quickest route, which is generally not the most direct route.

Quickest route? Wait! This is what Bernoulli was looking for! But before getting into that you need to understand a few principles.

You might have noticed that a spoon half-immersed in a glass of water appears bent at the surface. The underlying phenomenon that explains this is the refraction of the light. The speed of light depends on the medium through which it travels, and when light crosses a boundary between two media (plural form of medium), it gets refracted.

A spoon in a glass of water. Source

Fermat’s principle indicates that we can find the solution to the brachistochrone problem by imitating the behaviour of light. Light always finds the quickest path because it follows Snell’s Law of Refraction.

It is getting pretty heavy with all these principles, and things might be going over your head, right? So, let us try to understand the gist of these principles using a simple example:

Imagine that you and your friend went to a beach. Your friend is standing a bit further down the shore, and you want to reach him in the least amount of time. Now, a straight-line path would be the path of the shortest distance, but you know that you can run on sand much faster than you can swim in water. So, instead of a straight-line path, you would choose a path that increases your time on the sand, where you move faster, and decreases your distance in the water, where you move slower. 

Different path options for you to reach your friend. Source

The ratio of sines of the angles made with the normal (look in the figure below) equals the ratio of speeds in different media, in this case, sand and water. Ultimately, the final path that satisfies this condition will be the path of shortest time.

Refraction of light

Cycloid, the destination of the voyage

By utilizing the equations derived from Snell’s law and considering the velocity gained by the point mass traveling through the curve under gravity (the only force under consideration), Bernoulli ultimately arrived at a differential equation representing a curve known as the cycloid.

A cycloid is a curve traced by a point on the circumference of a circle as it rolls along a straight line without slipping.

Cycloid. Source

The left half arc of the inverted cycloid shown in the following figure, known as tautochrone, is identical to the path we are looking for. So we conclude that the solution to the brachistochrone problem is a cycloid.

(Also, check out this 4-minute video for a crisp animation!)

Inverted Cycloid. Source

Now, you likely understand the source of exhilaration on roller coasters or skating rinks – you're traversing a path that minimizes the time needed to cover the entire distance.

Comparison of time taken by a particle to travel along different paths. Source

Genesis of the Calculus of Variation

The Brachistochrone Problem holds great significance in the history of Mathematics and Physics. The array of solutions put forth by different scientists for the Brachistochrone Problem not only tackled Bernoulli's challenge but also laid the foundation for a new analytical method—variational calculus. This method is instrumental in studying dynamical systems within Classical Mechanics.

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Sources:

1. The Beauty in Mathematics - Data Genetics

2. The Brachistochrone - whistleralley.com

3. Brachistochrone curve - Wikipedia

4. The Brachistochrone - Vsauce

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