I am looking for a reference which contains a coordinate-free derivation of the Euler-Lagrange equation of the $p$-energy between Riemannian manifolds:
$$E(f)=\int_M \|df\|^p, f:M \to N.$$
(I only found treatments which use coordinates).
Is there such a reference? (For the classical case $p=2$, there are plenty).
I am writing a quick sketch using Anthony Carapetis's idea:
By the chain rule,
$$d(\|df\|^p)=d\big((\|df\|^2)^{\frac{p}{2}}\big)={\frac{p}{2}}(\|df\|^2)^{\frac{p}{2}-1}d(\|df\|^2)={\frac{p}{2}}\|df\|^{p-2}d(\|df\|^2) \tag{1}$$
So, given a variation field $V \in \Gamma(f^*TN)$, we get that (up to constants)
$$\frac{d}{dt}E(f_t)=\int_M \|df\|^{p-2}\langle df, \nabla V \rangle=\int_M \langle \|df\|^{p-2}df, \nabla V \rangle=\int_M \langle \delta(\|df\|^{p-2}df), V \rangle.$$
In fact, we can also compute "directly", without using the chain rule first:
$$d(\|df\|^p)=p\big(\|df\|^{p-1}d(\| df \|)\big),$$
and now (using the chain rule...)
$$ d(\| df \|) = d(\sqrt{\|df\|^2})=\frac{1}{2\sqrt{\|df\|^2}}d(\|df\|^2)=\frac{1}{2\|df\|}d(\|df\|^2),$$
So $$d(\|df\|^p)={\frac{p}{2}}\|df\|^{p-2}d(\|df\|^2),$$
just like in equation $(1)$.