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}

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