Second variation

The second variation is the quadratic term in a functional's expansion along $u+\epsilon \eta$; positivity on admissible variations is a local-minimum test.

The Jacobi equation is the linearization of an Euler-Lagrange equation about a stationary path. Its solutions describe infinitesimal one-parameter families of stationary paths.

A conjugate point marks a nonzero endpoint-vanishing Jacobi variation and the loss of positive definiteness of the second variation.

For a Jacobi field along a geodesic with nonpositive sectional curvature, the curvature convention with positive round-sphere curvature givesThus $|J{|}^2$ is nonnegative and convex. If it vanishes at both ends of a segment, convexity forces it to vanish everywhere. No nonzero Jacobi field can vanish at both ends, so the segment has no conjugate points. Only curvature of planes containing the geodesic tangent is needed, and completeness is unnecessary.

For $f(0)=f(T)=0$,with equality for a multiple of $\sin (\pi t/T)$.

If a continuously differentiable $L$-periodic function $f$ has mean zero, thenEquality holds exactly for $f(s)=A\cos (2\pi s/L)+B\sin (2\pi s/L)$.