Updated: May 31, 2021
I am in my fourth semester of going about analyzing proofs and various results arising from those incredible proofs, and the Brachistochrone is a part of one of those problems, essentially giving rise to calculus of variations. The Brachistochrone a geek word meaning shortest time, initially posed by one of the Bernoulli brothers, in this case Johann Bernoulli and is by far one of the most famous problems in the history of mathematics. Turns out that it was presented as a challenge “To the most acute mathematicians of the entire world.” (Which included Isaac Newton, who was old by then but hadn't lost his esoteric nature.)
Before, we get to the problem itself, I would like to share my thought as to why this problem and its solution is or might be profound. Well, one of the main reasons in my opinion being that it provides us with such a simple and elegant way of finding a connection between simple analytical geometry and essentially a law of nature, as in a physical a aspect present in the framework of our nature here on earth and the universe as a whole, if this seems confusing it'll be explicit to you, once you go through the problem and the solution.
Here's what Johann Bernoulli put out for his brethren,
"Given two points on a plane at different heights, what is the shape of the wire down which a bead will slide (without friction) under the influence of gravity so as to pass from the upper point to the lower point in the shortest amount of time?"
Now, well before such a problem was exclusively asserted, Galileo Galilei also had the same question pop in his head well before Bernoulli, in the 1600's. And to his credit he did do some relevant work to come to a conclusion to this problem but, it wasn't enough. He studied an objects motion under gravity, depicting that a body falling in space will traverse a distance proportional to the square of the time of descent. Using this law, he was able to compute the time of descent of an object falling along an inclined plane from point a A to point B, assuming that there is no effect of friction. Then the shortest path from A to B is a straight line, but is this path truly the one that will take the shortest time?
For instance, you could imagine the points A and B joined by two vectors from A to C, C to B having some angle between them, by its nature the body would fall following a straight line from A to C, and then a straight line from C to B, it would do so in less time than if it traversed a straight-line path from A to B?
Galileo believed that the answer to the brachistochrone problem was an arc of a circle, i.e., that the path A→C→B should be replaced by a circle passing through points A and B.
Well, not going to discuss this piece of math here but only keep the result in mind, so that we can move on to the other piece of math shown by Bernoulli himself, It turns out we found a better way, which actually minimizes time more than this arc of a circle. Before we get to that, one of the first things Johann thought about was the peculiar nature of the way in which light travels. In the 1600's, Fermat, showed us the principle of least time, in which he stated the way that light travels, whether bouncing off of a mirror or refracting from one medium or going through a lens, where light essentially bends. All the motion of light could be understood by saying that light takes the path that gets it from one medium to the other in the shortest time. Which definitely seems mystical, but as I said in the beginning, this is probably one of those big physical aspects embedded in the framework of nature, it could have been anything out of many things, could think of it as just one outcome from many and not get too philosophical about it.
Fermat's principle : "All the paths that light might choose to get from one point to another, it always chooses the path which takes the least amount of time. This principle is more general since it equally applies to both uniform and non-uniform media."
Johann, very intuitively assumed that instead of a sliding bead, it was light traveling through media of different index of refraction. Meaning that the light would go at different speeds as it successively went down like the bead.
Light bends when it goes from one medium into another, where its speed changes, consider the angle that it makes with a line perpendicular to the boundary between those two mediums. The sine of that angle divided by the speed of light stays constant as you move from one medium to the next, this is known as Snell's Law.
Johann Bernoulli proficiently used Snell's law, i.e., sin𝜃 over 𝑣 stays constant, and uses it to solve the Brachistochrone problem. When he thinks about what the bead, about what it might be going through as it's sliding down, he detects, that by the law of conservation of energy, as the bead slides down it covers some vertical distance and as a result has a transition of energy from potential energy to kinetic energy. As a consequence, the velocity that the bead will be proportional to the square root of the distance from the top, shown in the image below. The loss in potential energy is its mass times the gravitational constant times h, i.e., the distance from the top. When you set that equal to the kinetic energy, one-half times 𝑚𝑣 squared, and you simplify, the velocity 𝑣 will as a matter of fact end up being proportional to the square root of h.
As a result, considering the trajectory of light as it moves from one medium to the other from above, while instantaneously obeying Snell’s law, i.e., the ratio of sin𝜃 and v is always a constant, as we move from one medium to the next, then what will be the path, such that the tangent lines as shown above, are always instantaneously obeying Snell’s law?
At the end of it, the deduction that Johann came to conceive, was that the time-minimizing curve had the sine of the angle between the tangent line at that point and the vertical divided by the square root of the vertical distance between that point and the start of the curve, as shown in the figure above (Snell's law), will be some constant independent of the point that you chose. Johann Bernoulli saw this and recognized it as the differential equation for a cycloid. (Which at first glance isn't certain at all)
Above, is a desmos graph describing a cycloid, which is essentially the path traced by the point on the cicumference of a rolling wheel. This is the part where things get interesting, Mark levi a mathematician who is renounced for his intuitive ways, through which he uses principles of mechanics and, more generally, physics to solve all kinds of math problems. As he points out if you examine the geometry of a cycloid, through a few modifications at the right places, the principle of velocity over sin 𝜃 being constant ,i.e. Snell's law is embedded into the motion of the cycloid itself. Which in my opinion, is just splendid.
Here's the solution from his original article, a very simply and elegant explanation to how things theoretical/analytically can relate to actual physical aspects of nature.