The Brachistochrone Curve

Let $B$ to be a ball positioned at some point $A$. Considering that only gravitational force acts on $B$, what is the trajectory the ball should follow in order to reach a point $C$ 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 $n_1$ and $n_2$ constants related with the speed of light at mediums $M_1$ and $M_2$ (also called refraction indexes of mediums $M_1$ and $M_2$); by $\theta_1$ the entry ray’s angle with the plane $P$ separating $M_1$ and $M_2$; and by $\theta_2$ the out ray’s angle, the relation is written as:

$\displaystyle \frac{n_1}{n_2} = \frac{ \sin \theta_1 }{\sin \theta_2}$