The Brachistochrone Problem

The Brachistochrone Problem is an optimization problem and also one of the most famous problems in the history of mathematics. It was posed by Swiss mathematician Johann Bernoulli to the readers of Acta Eruditorum in June, 1696 as a challenge:

“I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.”

The problem he posed was the following:

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

Five mathematicians responded with solutions: Isaac Newton, Jakob Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital.
History records Newton as solving it the fastest. Newton found the problem in his mail, in a letter from Johann Bernoulli, when he arrived home from the mint at 4 p.m., and stayed up all night to solve it and mailed the solution by the next post. This story gives some idea of Newton's power, since Johann Bernoulli took two weeks to solve it.

Johann Bernoulli was not the first to consider the brachistochrone problem. Galileo in 1638 had studied the problem (without the benefit of Calculus) in his famous work Discourse on two new sciences. His version of the problem was first to find the straight line from a point A to the point on a vertical line which it would reach the quickest. He correctly calculated that such a line from A to the vertical line would be at an angle of 45 reaching the required vertical line at B say.
He calculated the time taken for the point to move from A to B in a straight line, then he showed that the point would reach B more quickly if it travelled along the two line segments AC followed by CB where C is a point on an arc of a circle.
Although Galileo was perfectly correct in this, he then made an error when he next argued that the path of quickest descent from A to B would be an arc of a circle - an incorrect deduction.

The animation below visualizes the problem of brachistochrone. The path that describes this curve of fastest descent is given the name Brachistochrone curve (after the Greek for shortest 'brachistos' and time 'chronos') and it is a cycloid.

The problem can be solved with the tools from the calculus of variations and optimal control.
To calculate the optimal path requires minimizing a function that minimizes some other variables. This is the calculus of variations. There are many excellent papers available that walk through the process.

Animation source>> http://www.maa.org/book/export/html/438626