# The Brachistochrone Problem

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The Brachistochrone Problem As Archimedes stated, “The shortest distance between two points is a straight line.” (Archimedes Quotes) The Brachistochrone problem is based on the paradox of this assertion. Consider two points P and Q where point P is situated at some distance at a higher level from point Q. So, when a ball moves from P to Q, under the influence of gravity and no other force, the straight line that joins P and Q must be the fastest way to reach Q from P. Surprisingly, this appears to be false because the fastest path from P to Q is given by a cycloid curve. Figure 1. “A cycloid is the locus of a point on a rim of a circle (with a certain radius) rolling along a straight line.” (Wolfram Math World) Figure 2. Despite knowing…show more content…
L (x,y,y^')= √((1+(y^' )^2)/2gy) (12) So, finding the optimum value of T using the Euler-Lagrange equation is the problem under calculus of variations. Let suppose M is defined by an integral in the form: M=∫_x1^x2▒L(x,y,y^' )dx (13) Here, M is called the functional and has a constant extreme value if the Euler-Lagrange equation is satisfied. The Euler-Lagrange equation is: ∂L/∂y-d/dx (∂L/(∂y^' ))=0 (14) *Note: The derivation of the Euler-Lagrange equation will not be a part of this exploration. So, to solve Euler-Lagrange equation, we might substitute the value of L from (12) to equation (14). Having said that, since L (x,y,y^')= √((1+(y^' )^2)/2gy) does not contain “x” explicitly and L is the function of y and y^'only, equation (14) cannot be