# Gradient of the energy functional in $H^{1,2}$-norm

I have to use estimates for the gradient of the energy functional on the free loop space of a fixed compact manifold $Q$. As such, one considers $H^{1,2}$-maps of the circle into $Q$. The energy functional is given by $\mathcal{E}:H^{1,2}(S^1,Q)\rightarrow IR ,\quad \gamma\mapsto \frac{1}{2}\int_0^1\|\dot{\gamma}(t)\|^2dt$. Is there a neat formula for the gradient w.r.t. the $H^{1,2}$ inner product?

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This is a straightforward computation that you should do yourself. But what does "neat" mean here? –  Deane Yang Sep 7 '10 at 15:06
The computation is straightforward w.r.t. $L^2$, not $H^{1,2}$. –  Orbicular Sep 7 '10 at 15:20
I only know of a variational characterization, i.e. in terms of an ODE, but no "explicit" form. –  Orbicular Sep 7 '10 at 15:21
Also, isn't the energy functional equal to $\int |\dot\gamma|^2$ and not $\int |\dot\gamma|$? –  Deane Yang Sep 7 '10 at 15:27
@Deane: It could be that a double vertical bar means the square of the norm... I have come across this notation before. –  José Figueroa-O'Farrill Sep 7 '10 at 15:39