This is one of the classic mathematical problems whose solution led to the discovery of the calculus of variation – now very useful in the field of engineering and physics. Finding its solution for the first time was not easy. It took the great Bernoulli brothers, Newton, Leibniz, et al. to accomplish it. All of them are all-time mathematical greats.
Our intuition would most likely tell us that any object allowed to fall under the influence of gravity alone would take the natural straight line course despite the object being constrained to travel as quickly as possible – travel of least time. And we are inclined to come up with such a conclusion for it’s the seeming natural appearance of events – the Aristotelian method of deducting scientific conclusions.
However, as the annals of history would unfold, humankind is a whole story of gradual progress across fields of endeavor; albeit, scientific progress was greatly hampered by the Church. Fraught with intellectual conscience and honesty, Galileo courageously braved the mostly erroneous scientific dogmas of the Church and his effort paved the advent of scientific revolution and enlightenment.Thereon, European scientists and mathematicians raced their way to scientific fame by outdoing one another in trying to come up with original mathematical and scientific discoveries. Providing solutions to challenge problems in mathematics once became the crux of mathematical efforts in intellectual Europe.Example of which is the now famous brachistocrone (German for mind-boggling) problem. It requires finding the actual geometry of the path that allows any object to travel from a designated point to another in the quickest possible way. The object is free from other forces except gravitation.
The great Swiss mathematician Johann Bernoulli posed the problem for all mathematicians of Europe to solve. Bernoulli, though, was the first person to solve it, but he was not the first to point it out. Galileo himself investigated the problem and obtained the solution experimentally but fall short from being exactly correct. Using the newly discovered mathematics of calculus and the available physics of free-falling bodies, many brilliant and hard working mathematicians found the correct solution. Gottfried Leibniz, Jacob Bernoulli (brother of Johann), and Isaac Newton all arrived at the same truth – any moving object under the action of gravity should take a curved path – part of an inverted cycloid (blue curve shown below) – if it is to travel from one point to another within the shortest time possible.
Interestingly, Johann Bernoulli allowed six months for anyone to come up with the solution. Sir Isaac Newton, the architect of classical physics and who shared credit with Leibniz for the discovery of calculus solved it overnight. And further investigation of this classic problem led to the formulation of the calculus of variation – the modern mathematical tool used to solve problems involving maxima and minima situations. By which the problem presented above can now be solved with much greater ease.
An exact mathematical proof is found here