Some "Eurekas" of Mathematics - Part Six

A natural question that has arisen in mathematics is one that involves that fastest descending path. For example, say you need to ride your bicycle down into town every day, but have various paths to take. For example, say one path is very steep at first, then levels out for the rest of the time to the town, while another path keeps a constant slope the entire descent into town. As you may see by now, there are a number of different paths which could possibly be created to get you into town the fastest (without even needing to pedal the bicycle).

This is where the brachistochrone problem arises. It was first proposed in the late 17th century by Johann Bernoulli, shortly after the discovery of calculus. Bernoulli, who is believed already had his solution, asked the mathematics community:

"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."

Bernoulli had the intention of flattering the mathematics community. Various mathematicians submitted solutions over the next few months, including Bernoulli's brother and Isaac Newton who sat down and solved the problem in about 12 hours. In the end, Johann Bernoulli's solution ended up containing an error, so he tried to claim his brother's correct solution as his own.

So what was the solution to Bernoulli's problem? As it turns out, the fastest curve follows what is known as the brachistochrone curve. The brachistochrone curve is a type of cycloid. That is, a curve which is traced by a point on a circle as the circle rolls.

The height difference between the starting and ending points will determine how much of the cycloid is traced.

So as it turns out, the fastest path down into town wouldn't follow a constant slope, nor would it have a steeper slope at the beginning to build up more momentum from gravity. In fact, the fastest route would require descent followed by an ascent at the end, just as seen in the photo above. (The less of a descent, relative to total distance, the more of an uphill in the second half of the curve.)

As was mentioned in the introduction, this problem was proposed shortly after the creation of calculus. This is no coincidence as calculus is needed to derive the solution. In fact, this problem is often considered the start of a branch of mathematics known as calculus of variations. Unfortunately, deriving the solution here on Jetpunk wouldn't be easy and would probably be a total waste of time.

If you're interested in seeing the solution get testing in real life, check out this video.