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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).

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  • $\begingroup$ You should be able to follow the same method as the classical case using the chain rule on $(\|df\|^2)^{p/2}$ - are you having trouble working this out yourself, or you do just want something nice to cite? $\endgroup$ – Anthony Carapetis Jun 13 '17 at 9:59
  • $\begingroup$ Thanks. Embarassingly, I indeed had trouble with it:) I guess I was too tired when trying to compute this, like you said it's very easy. (I thought deleting the question, but someone might find it useful in the future, so I left it as it is). I really should learn to wait a bit more before asking.... $\endgroup$ – Asaf Shachar Jun 13 '17 at 10:49
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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 $(1)$.

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