Jacobi equation

Remarks

For $h \in C^1([a, b])$ with $h(a) = h(b) = 0$, the second variation of $J$ at $y_0$ equals $\int_a^b (P h'^2 + Q h^2) \, dx$. The Jacobi equation is the Euler–Lagrange equation of this quadratic functional. It is also the linearisation of the Euler–Lagrange equation of $J$ at $y_0$. If $y(x, c)$ is a smooth family of extremals with $y(x, 0) = y_0(x)$, then $h = \partial y / \partial c$ at $c = 0$ solves it.

Example: for $L = p^2 - y^2$ and $y_0 = 0$, one has $P = 2$ and $Q = -2$. The Jacobi equation is $h'' + h = 0$; the function $h = \sin x$ satisfies it, since $h'' = -\sin x = -h$.

Non-example: for the same $L$, the function $h = x$ does not solve it, since $-2h'' - 2h = -2x \ne 0$ for $x \ne 0$. It does solve the Jacobi equation $h'' = 0$ of $L = p^2$, whose extremals are straight lines.

When $P > 0$ on $[a, b]$, the equation has a unique solution for given $h(a)$ and $h'(a)$. For curves $y \colon [a, b] \to \R^n$ the same construction gives a linear system with matrices $P$ and $Q$.