Abstract:  The variational principle is introduced, and its Euler-Lagrange equation for the minimum of variational problems in the setting of one dimension is described. Then, the corresponding specific expression of these Euler-Lagrange equations for taking a geodesic line, geodesic curve, Brachistochrone and isoperimetric problems are deduced as examples, respectively. The solutions of the corresponding original problems are obtained and the relevant generalizations are given.

Key words: variational principle, calculus of variations, Euler-Lagrange equation, extremum function