Let to be a ball positioned at some point . Considering that only gravitational force acts on , what is the trajectory the ball should follow in order to reach a point in the shortest time? The solution of this problem is known as the brachistochrone curve.
Although similar, the brachistochrone problem is different from the tautochrone problem. The amusing fact is that the solutions of both problems are the same. The brilliant argument exposed next is due to Johann Bernoulli.
The principle of least time
Light has different speeds at different mediums. That’s why we perceive the phenomenon of refraction when observing objects underwater while ourselves stand out of it. Willebrord Snellius was the first to give a relation between the speed of light in different mediums and its angle of refraction. By denoting and constants related with the speed of light at mediums and (also called refraction indexes of mediums and ); by the entry ray’s angle with the plane separating and ; and by the out ray’s angle, the relation is written as:
Snellius probably based its relation on experimental evidence, but we can derive it from the principle of least time. It was argued by Pierre de Fermat that nature behaves in an optimum way. In particular, the light would follow the path of least time between any two points and . In the following, we are going to use Fermat’s principle to derive Snell’s law.
Assume points and are separated from each other by a height of and a length of . Moreover, assume the plane separating mediums and is at a height from . It will be sufficient to discover the point , distant of a length from , in which the entry ray touches the plane . The time light will take to travel from to is denoted and time light takes from travel from to is denoted . We can easily compute and :
The value that minimizes is then
In other words, the ratio is constant. We are going to call this constant .
If the path traveled by light is the one of least time, maybe we can use this fact to derive the brachistochrone curve. That was precisely the idea of Johann Bernoulli. Imagine that a ball traveling from points to passes through many different mediums. In order to follow the path of least time, the ball’s trajectory should suffer the same changes of orientation that light would suffer by Snell’s law. If we consider thinner and thinner mediums we hope that we can find the brachistochrone curve using calculus:
By denoting we recover the equation of the tautochrone curve. Surprisingly, the tautochrone and the brachistochrone curves are the same: an arch of a cycloid!