= Weierstrass necessary condition
{c}
{title2=$E(x,u,u',v)\geq0$}
At a <strong local minimizer of a variational integral>, almost every reference slope $u'(x)$ is a <point of convexity> of $v\mapsto f(x,u(x),v)$. Equivalently its <Weierstrass excess function> is nonnegative for every trial slope. Uniformly small variations can have large derivatives, which is why this condition tests all trial slopes rather than only nearby ones. It is a necessary condition along the minimizing curve; the <strengthened Weierstrass condition> additionally controls a neighborhood of that curve and its slopes.
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