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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
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Is there any reason to believe that there should be an explicit formula? – Deane Yang Sep 8 '10 at 1:32

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