Second variation

Summary

The second variation of a functional at a function $y_0$ is the second derivative of $\varepsilon \mapsto J(y_0 + \varepsilon h)$ at $\varepsilon = 0$. It plays the role of the second derivative test of ordinary calculus. At a minimiser the first variation vanishes and the second variation is nonnegative in every direction $h$. For integral functionals it is a quadratic functional in $h$, and its study leads to the Legendre condition, the Jacobi equation and conjugate points.