Euler-Lagrangeequation is an important equation in the variational method. It provides a method to find the stationary value of functional, and its initial idea is that "the derivative extreme point must be a stable point (critical point)" in elementary calculus theory.

When the energy functional contains differential, the proof process is deduced by variational method. Simply put, assuming that the current function (that is, the real solution) is known, this solution will inevitably make the energy functional take the global minimum.

when finding the maximum and minimum values of a function, the analysis of small changes near a solution gives an approximation of the first order. It can't tell whether it finds a maximum or a minimum (or neither).