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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