An identity in calculus of variations discovered in 1868 by Beltrami. The Euler-Lagrange differential equation is (df)/(dy) - d/(d x)((df)/(dy_x)) = 0. Now, examine the derivative of f with respect to x (d f)/(d x) = (df)/(dy) y_x + (df)/(dy_x) y_(x x) + (df)/(dx). Solving for the df/dy term gives (df)/(dy) y_x = (d f)/(d x) - (df)/(dy_x) y_(x x) - (df)/(dx). Now, multiplying (-3) by y_x gives y_x (df)/(dy) - y_x d/(d x)((df)/(dy_x)) = 0.

brachistochrone problem | calculus of variations | Euler-Lagrange differential equation | surface of revolution