Evolving Brachystochrone Through Simulated Evolution

In 1696, the famous mathematician of the Bernoulli family, Jean Bernoulli, challenged his peers posing an interesting physical problem:

"Given two points P and Q in a vertical plane and taking into account the straight line holding them is not vertical neither horizontal, what is the curve connecting them such that a particle starting from the higher point P with no initial velocity (V₀ = 0) and sliding down through that line without friction, under the influence of gravity, takes the smallest amount of time to reach the lower point Q?"

A simple and crude schematic of that problem may be seen below:



That problem was presented in the June issue of the famous mathematical journal Acta Eruditorum.

Five other mathematicians replied with a solution and only four of them were published in the next year's May issue of that journal. The whole group was:



The solution curve was named brachistochrone (from the Greek words βραχίστος, brachistos - the shortest; and χρόνος, chronos - time) by Leibniz.



Jean's solution was valid only under certain conditions what made him later propose another and harder version of the same problem, resulting in what became known as calculus of varitations.

This problem itself is an optimization one, in this case optimization of time. So... what about using evolutionary computation to solve the same problem some 320 years later? You may not be of such genius stock as those guys above, but evolutionary computation makes easier to you solve the very same problem! Enjoy it!