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).

  • $\begingroup$ 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

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 $(1)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.