Jacobi equation

Summary

The Jacobi equation of a variational problem is the linear second-order differential equation $-(P h')' + Q h = 0$ obtained from the second variation along an extremal. It is the Euler–Lagrange equation of the quadratic functional $\int (P h'^2 + Q h^2) \, dx$, and also the linearisation of the Euler–Lagrange equation along the extremal. Its solutions describe how nearby extremals spread apart, and their zeros define conjugate points. For the length functional on a Riemannian manifold it becomes the equation of a Jacobi field.