Very basic question, but we have recently gone over the topic of geodesics in class and I was wondering whether it was permissible to apply Euler-Lagrange to find the extremal of say $ \int \text d s^2 $ instead of the ordinary arc length.
I've tried the case of a straight line and it seems to produce the same result but I'm not sure if this principle is applicable in general (primarily just to eliminate clutter from repeated differentiation of square roots).
If it is part of a more general idea, what are the conditions under which it is valid and in which case it is equivalent to apply this form (e.g. would it also be valid to consider $ \int \text d s^3 $ or $ \int \text d s^4 $, or perhaps a variety of other functions of the original integrand under consideration )
It is not just possible, but desirable. The reason is that the critical points of the arclength $L$, viewed as a functional on, say, $C^\infty([0,1],M)$, are highly degenerate: if $\gamma \colon [0,1] \to M$ is a (local) length minimizer then so is any reparametrization of $\gamma$, so $\gamma$ belongs to an infinite dimensional critical locus homeomorphic to the diffeomorphism group of $[0,1]$.
On the other hand, by the Cauchy-Schwarz inequality the energy function $E(\gamma) = \int_\gamma ds^2$ satisfies the inequality $L(\gamma)^2 \leq 2 E(\gamma)$ with equality if and only if $\gamma$ has constant speed. This implies that the critical points of $E$ are precisely the constant speed parameterizations of locally length minimizing curves, so the critical locus near such a curve is just a line. This means the behavior of $E$ near its critical loci is much easier to analyze. And it doesn't hurt that the Euler-Lagrange equations of $E$ are a lot easier to write down and solve.