Brahistochrone Research Paper

The Brachistochrone

Name Manjunath O K

Registration number 2017MPH21

Date of presentation 01-02-2018

The word Brachistochrone originates from the Greek word for ‘shortest time’. The brachistochrone curve was originally a mathematical problem posed by Swiss mathematician Johann Bernoulli in June 1696, and the problem is this: If a body is released from rest at a point A, what is the path of quickest descent for the body to take from point A to a lower point B under only the influence of gravity? He challenged the scientific community to come up with solutions. At the end, five mathematicians responded with solutions: Newton, Jacob Bernoulli (Johann's brother), Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital.

Jacob Bernoulli posed isoperimetric problems to Johann Bernoulli and a bitter dispute arose between the two brothers on these problems which Varignon also became involved in. It was an unpleasant incident, but one of great value to science for the problems being argued about led directly to the founding of the calculus of variations.

The methods which the brothers developed to solve the challenge problems they were tossing at each other were put in a general setting by Euler .He generalized the problems studied by the Bernoulli brothers but retains the geometrical approach developed by Johann Bernoulli to solve them. He found what has now come to be known as the Euler-Lagrange differential equation for a function of the maximising or minimising function and its

Functionals are often expressed as definite integrals involving functions and their derivatives. Today variational calculus is found to be of extensive use in many branches of physics including general relativity and quantum mechanics. It all began with the Brachistochrone and what the world saw was the ego between two mathematicians contributing the scientific community a brand new tool to attack many complex scientific