# Find a functional for a given variation

I need to find a function $L(\phi)$ such that the functional $G=\int_{\Gamma} L(\phi) ds$ has a variation $$\frac{e^{-\phi(s)}}{\int_{\Gamma} e^{-\phi(t)} dt },$$ where $\Gamma$ is a 3-D surface. In other words $L(\phi)$ shall be such that $$\frac{d L(\phi)}{d \phi} = \frac{e^{-\phi(s)}}{\int_{\Gamma} e^{-\phi(t)} dt },$$ and I was stucked here. What will $L(\phi)$ be? Thanks.

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Something is missing. The variation should also depend on the direction: $\frac{d}{dt}L(\phi+t\psi)\Big|_{t=0}$. – Pietro Majer Jul 4 '12 at 6:10
Thanks for the comments. You are certainly correct here. It actually follows that the variation of a functional $G(u) = \int L(x,u,u')dx$ can be defined to be $$\frac{\partial L}{\partial u} + \frac{d}{dx} \frac{\partial L}{\partial u'}.$$ – paullastCO Jul 5 '12 at 4:27