fa.functional analysis - 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
$mathcal{E}:H^{1,2}(S^1,Q)rightarrow IR ,quad gammamapsto frac{1}{2}int_0^1|dot{gamma}(t)|^2dt$. Is there a neat formula for the gradient w.r.t. the $H^{1,2}$ inner product?