Why can the Euler-Lagrange equation be used to find the extremum of a functional?

Question

The Euler-Lagrange equation is a differential equation. A functional is a function from some vector space to a real number. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation. Why exactly can the Euler-Lagrange equation be used to find the minimum or maximum of a functional?

Answer 1

If you want to find the maximum or minimum of a differentiable function $f:(a,b) \to \Bbb R$, what do you do? You look for solutions of $f'(x) = 0$ (critical points) and then see whether they're local maximum or minimum using techniques from single-variable calculus, right? A "first derivative test".

This is precisely what the Euler-Lagrange equations are: a "first-derivative test" for finding functions which extremize a given functional. Solving the E-L equations will give you the critical points of the given functional, and then you can decide whether they're maximum or minimum (local or global) by geometric considerations or more elaborate "second derivative tests".

That is to say, $f'(x) = 0$ is the "same" as $\delta J[y] = 0$. For the actual proofs, there is a lot of books you can study (I like Gelfand's Calculus of Variations, van Brunt's The Calculus of Variations and Kot's A First Course in the Calculus of Variations).