This animation is about one of the most significant problems in the history of mathematics: The Brachistochrone Challenge:
If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?
Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.
Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics,Calculus of Variations, had to be invented to deal with such problems. Today, calculus of variations is vital in Quantum Mechanics and other fields.Reference:http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Brachistochrone.html
http://www.hep.caltech.edu/~fcp/math/variationalCalculus/variationalCalculus.pdf
Animation via imgur
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