= First integral for a variational integrand independent of the function value
{title2=$f_v(x,u'(x))=c$}
If a smooth <functional> has integrand $f(x,v)$ independent of the <function> value, a weak local minimizer satisfies the displayed first <integral> <almost everywhere>. Its <first variation> gives $\int f_v(x,u')\varphi'=0$ for every compactly supported <test function>. Thus the <distributional derivative> of $f_v(x,u')$ vanishes, making it constant on the interval. Equivalently, any smooth compactly supported <function> with <integral> zero is the <derivative> of such a <test function>, so the first-variation identity says that $f_v(x,u')$ agrees with a single constant against all tests. This reduces the <Euler-Lagrange equation> from a second-order equation to an algebraic relation in $u'$.
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