Weierstrass excess function

Summary

The Weierstrass excess function of a Lagrangian $L(x, y, p)$ is $E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p) \cdot L_p(x, y, p)$. It measures how far the graph of $q \mapsto L(x, y, q)$ lies above its tangent at $q = p$. It gives the Weierstrass necessary condition for a strong minimum, and through Hilbert's invariant integral it gives the classical sufficient condition for a strong minimum.