The Fastest Path Is Not a Straight Line

The Brachistochrone Problem

Let's say someone beings drafting ideas for a new roller coaster ride, and with a drop distance calculated, they ask you to figure out how to get the coaster from the highest point to the lowest point as quickly as possible for maximum thrill. How would you design it? If I hadn't just freely given the answer in the title, would you have assumed a straight line from A to B is the correct shape of the drop? If not, what would you have guessed? The arc of a circle? An exponential decay curve? Half of a parabola?

From what I learned, Galileo Galilei was the first person to really investigate "the question of fastest descent", called the brachistochrone problem (meaning shortest time), and claimed that the arc of a circle was the best shape. He later said that he was probably wrong, and a "higher science" was necessary to figure out which one truly minimized travel time. Years later, Johann Bernoulli issued a challenge solve this problem to fellow mathematicians, specifically targeting Isaac Newton, as a way to prove his own competence. Newton did end up solving it, but so did Bernoulli—and his solution was so beautifully creative. Instead of turning to mechanics, Bernoulli, weirdly enough, turned to optics for a solution. 

Light "Knows" How to Optimize Time

For years, mathematicians knew that light refracted in water, but didn't know why. Really, why should it? Why doesn't light just continue forward in a straight line? It seems counterintuitive that light should deviate from the "path of least resistance", so-to-speak. The solution, at least to me, is mind-boggling. It turns out, as French mathematician Pierre de Fermat discovered, that light isn't taking the shortest path; light is taking the quickest path. Since the speed of light can't move as fast in water, it bends to accommodate and optimize the time necessary to get from point A to point B. 

Here's an easier example to picture: you're standing on the shore—sea to your right and sand to your left—and your friend throws a frisbee to you. Your friend is a terrible throw, and the frisbee lands in front of you and far into the deep water (so it is diagonal to you). How are you going to get the frisbee? Most people would probably just jump in the water and get to the frisbee in a straight line, but you probably run faster on the sand than you can swim, so ideally you should run for some distance, and then swim the rest of the way to the frisbee. This is an optimization problem that I did in my first calculus class, and apparently some dogs can optimize this very problem nearly perfectly. 

It's easy to assume that once a photon is released, it shouldn't change its trajectory for any reason. But somehow, it does. This has many deep implications and one can quickly go down a rabbit hole of trying to understand how this principle relates to Einstein's theories of relativity and the principle of least action upon which they are founded, but I digress. Back to the original topic.

Bernoulli Solves the Problem

Knowing that light will change its trajectory in different objects, Bernoulli so cleverly decided to treat the rolling object as a light particle moving through an infinite number of media with increasing indices of refraction, and then integrating the path. Doing so, he found that the optimal path was a type of cycloid. If you haven't heard of a cycloid, you can think like this: if you were to draw a red dot on a bicycle wheel and roll it across the XY plane, the resultant curve would be a cycloid. It also just looks like a vertically shifted sine/cosine wave, because it is.

This curve has another unique property—no matter where an object begins on the curve, the time it takes to reach point B is the same. In other words, if you had a brachistochrone marble ramp and dropped two marbles at different spots along the curve at the same time, they would both arrive at the bottom simultaneously. 

That's all I have time for today, but I hope you learned something. It was very fascinating to me! Watch the video below that explains this story much better than I ever could: