Let $a < b$, let $U \subset \mathbb{R}^n$ be open, let $\mathcal{O} \subset \mathbb{R}\times U\times\mathbb{R}^n$ be an [open set](/page/Open%20Set) containing

\begin{align*} \{(x,y_0(x),y_0'(x)) : x \in [a,b]\}, \end{align*}

and let $L: \mathcal{O} \to \mathbb{R}$ be a $C^3$ Lagrangian. Let $y_0 \in C^2([a,b];U)$ be an extremal of the Euler--Lagrange equation

\begin{align*} \frac{d}{dx}\partial_v L(x,y_0(x),y_0'(x)) = \partial_y L(x,y_0(x),y_0'(x)). \end{align*}

Assume the strengthened Legendre condition holds along $y_0$: for every $x \in [a,b]$, the symmetric matrix $L_{vv}(x,y_0(x),y_0'(x)) \in \mathbb{R}^{n \times n}$ is positive definite.

Let $Z: [a,b] \to \mathbb{R}^{n \times n}$ be the Jacobi matrix along $y_0$ generated by varying the initial velocity at the fixed point $(a,y_0(a))$. Equivalently, the columns of $Z$ solve the linearized Euler--Lagrange equation

\begin{align*} \frac{d}{dx}\bigl(L_{vy}(x,y_0(x),y_0'(x))\xi(x)+L_{vv}(x,y_0(x),y_0'(x))\xi'(x)\bigr)=L_{yy}(x,y_0(x),y_0'(x))\xi(x)+L_{yv}(x,y_0(x),y_0'(x))\xi'(x) \end{align*}

with initial conditions $Z(a)=0$ and $Z'(a)=I_n$. Assume that $a$ has no conjugate point along $y_0$ in $(a,b]$, meaning $\det Z(x) \neq 0$ for every $x \in (a,b]$.

Then, for every $\delta \in (0,b-a)$, there exist a relatively open neighbourhood $\mathcal{N}_\delta \subset [a+\delta,b]\times U$ of the graph segment

\begin{align*} \Gamma_{y_0,\delta} := \{(x,y_0(x)) : x \in [a+\delta,b]\} \end{align*}

and a $C^1$ map $P_\delta: \mathcal{N}_\delta \to \mathbb{R}^n$ such that $P_\delta(x,y_0(x)) = y_0'(x)$ for all $x \in [a+\delta,b]$, and every integral curve $y$ of the first-order system $y'(x) = P_\delta(x,y(x))$ is an extremal of $L$. Thus $P_\delta$ is a local central field on every compact subarc separated from the initial point $a$.