Verification of the veracity of brachistochrone curve and evolution of optimal control

An evocative solution for the brachistochrone problem is presented by Euler-Lagrange equation. The solution of brachistochrone problem by variational approach i.e. Euler-Lagrange equation leads to the curve of a cycloid also known as brachistochrone curve. In this paper the exactness of the brachistochrone curve is verified through simulated studies in MATLAB, taking into cognizance different other curves viz. straight line and parabola along with cycloid. By computing the time taken by the bead in traversing the path between two points under gravity along all the curves considered, it is seen after comparison that the time taken along the cycloid is the least.