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Brachistochrone

# Brachistochrone - Brachistochrone Challenge(Birth of...

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Brachistochrone Challenge (Birth of Calculus of Variations) Simon Stevinus (1548-1620) Dutch - father of statics Galileo Galilei (1564-1642) Italian - father of dynamics Isaac Newton (1642-1727) Gottfried Wilhelm Leibnitz (1646-1716) John Bernoulli (1667-1748) In June, 1696, John Bernoulli challenged the following problem before the scholars of his time: Given are two points A and B (B is not directly below A) in a vertical plane. A movable particle m is to descend from A to B without friction following the law of gravity (natural fall). Determine the path of m to reach B from A in the shortest time. 1

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Solution: Suppose the points A and B lie in the xy plane, the y axis directed vertically downward, and the x axis horizontal, with passage from A to B marked by an increase in x. Let the extremizing path have the equation ( ) y y x = . It is assumed that the initial speed v 1 of the particle to be given in the statement of the problem. Let the points A and B have the coordinates ( ) , x y 1 1 and ( ) , x y 2 2 , respectively, so that ( ) y x y = 1 1 and ( ) y x y = 2 2 . Since the speed along the curve is given by ( ) / v ds dt = , the total time of descent ( ) x x x x x x y dx ds I v v = = ¢ + = = ò ò 2 2 1 1 2 1 . where: ( ) ( ) dy ds dx dy dx y dx dx æ ö ÷ ç ¢ ÷ = + = + = + ç ÷ ç ÷ ç è ø 2 2 2 2 2 1 1 Assuming frictionless descent, for simplicity, of the particle of mass m with a constant gravitational acceleration g, the velocity v of the natural fall of the particle can be computed in terms of coordinates by invoking the principle of conservation of energy, i.e., a decrease of potential energy is equal to an increase of kinetic energy. Hence, ( ) mv mv mg y y - = - 2 2 1 1 1 1 2 2 where: v gh g y y = = - 0 2 2 and v g y y = - 1 1 0 2 v y y g = - Þ 2 1 1 0 2 . It is clear that y y - 1 0 is the vertical distance through which the particle must descend from rest to achieve the speed v 1 , and if v = 1 0 , then y y = 0 1 . Thus, the time of descent is 2
x x y I dx g y y ¢ + = - ò 2 1 2 0 1 1 2 . The problem becomes simply that of choosing a function ( ) y y x = which minimizes the integral. This problem (referred to as “Brachistochrone” derived from the Greek brachistos , shortest, and chronos , time) is considered the birth of a mathematical branch,

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Brachistochrone - Brachistochrone Challenge(Birth of...

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