I have a very basic question in the calculus of variations:

Suppose I want to minimize the functional

$$A[r, r'] = \int_\Omega L(r, r') dx $$

When is it possible to say that extremals of $A$ agree with extremals of

$$\tilde{A}[r, r'] = \int_\Omega \left( L(r, r') \right)^2 dx $$

Assume that $L(r, r') \geq 0$. Note that $r: \mathbb{R}^{n - 1} \rightarrow \mathbb{R}^n$ (e.g. $r$ as a parameterization of a minimal surface), $r'$ is really a matrix, and $L(r, r')$ is a real number.

Any references for these types of issues would also be great.