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For the following variational problem I have been told to show the Euler-Lagrange equation is identically zero.

$$L[u]:= \int_a^b \sin(u)u_x\,\mathrm dx $$

I have found it to be

$u_x\cos(u)-\sin(u)u_{xx}=0.$

Is this correct? And if so, does this always equal $0$?

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    I think your second term is wrong, it should be something like $\frac{d}{dx} \sin(u)$, which will end up canceling the first term.2012-10-21

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