Let to be a ball positioned at some point . Considering that only the gravitational force acts on the ball, 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 curve has a different description than the tautochrone curve. The amusing fact is that they are, indeed, 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 ); the entry ray’s angle with the plane separating and by ; and the out ray’s angle, the relation is written as: