## 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?

-
 This is a straightforward computation that you should do yourself. But what does "neat" mean here? – Deane Yang Sep 7 2010 at 15:06 The computation is straightforward w.r.t. $L^2$, not $H^{1,2}$. – Orbicular Sep 7 2010 at 15:20 I only know of a variational characterization, i.e. in terms of an ODE, but no "explicit" form. – Orbicular Sep 7 2010 at 15:21 Also, isn't the energy functional equal to $\int |\dot\gamma|^2$ and not $\int |\dot\gamma|$? – Deane Yang Sep 7 2010 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 2010 at 15:39