An optimization example: The brachistochrone problem

In the teaching of optimization methods such as in Control courses, a frequently used introductory example is the classical brachistochrone problem. The Calculus of Variations solution is usually obtained by introducing a new parameter to solve the nonlinear differential equation. This letter presents an alternative method which introduces as a parameter the instantaneous angular direction of the falling particle and obtains a solution in which a nonlinear differential equation does not arise.