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?
