Let $U \subset \mathbb{R}$ be an open interval, let $a,b \in U$ with $a < b$, and let $L \in C^2(U \times \mathbb{R} \times \mathbb{R})$. Let $y \in C^2([a,b])$ be stationary for the endpoint-moving functional whose unperturbed value is

\begin{align*} I[y] = \int_{[a,b]} L(x,y(x),y'(x))\,d\mathcal{L}^1(x) \end{align*}

among admissible endpoint-moving variations with fixed left endpoint and right endpoint constrained to a regular $C^1$ curve $\Gamma \subset \mathbb{R}^2$.

More precisely, assume each admissible variation is represented, for $|\varepsilon|$ sufficiently small, by a pair $(b_\varepsilon,Y_\varepsilon)$ such that $\varepsilon \mapsto b_\varepsilon$ is $C^1$, $b_\varepsilon \in U$, $a < b_\varepsilon$, $b_0=b$, $Y_\varepsilon \in C^2(U)$, $Y_0|_{[a,b]}=y$, and the map $(\varepsilon,x) \mapsto Y_\varepsilon(x)$ is $C^2$ on $(-\varepsilon_0,\varepsilon_0) \times U$. Assume also that the left endpoint is fixed,

\begin{align*} Y_\varepsilon(a)=y(a), \end{align*}

and the right endpoint satisfies

\begin{align*} (b_\varepsilon,Y_\varepsilon(b_\varepsilon)) \in \Gamma. \end{align*}

The value of the functional on this variation is

\begin{align*} I(\varepsilon)=\int_{[a,b_\varepsilon]} L(x,Y_\varepsilon(x),\partial_xY_\varepsilon(x))\,d\mathcal{L}^1(x). \end{align*}

Stationarity means that $I'(0)=0$ for every admissible variation.

Assume the admissible class contains all fixed-endpoint interior variations: for every $\phi \in C_c^1((a,b))$, the family $b_\varepsilon=b$ and $Y_\varepsilon=y+\varepsilon\phi$ is admissible for $|\varepsilon|$ sufficiently small. For an admissible variation, define its endpoint velocity by

\begin{align*} \tau=\frac{d}{d\varepsilon}\Big|_{\varepsilon=0}(b_\varepsilon,Y_\varepsilon(b_\varepsilon)) \in T_{(b,y(b))}\Gamma. \end{align*}

Assume conversely that every vector in $T_{(b,y(b))}\Gamma$ occurs as the endpoint velocity of some admissible variation. Then, for every $\tau \in T_{(b,y(b))}\Gamma$,

\begin{align*} \left(L(b,y(b),y'(b))-y'(b)\partial_{y'}L(b,y(b),y'(b)),\partial_{y'}L(b,y(b),y'(b))\right)\cdot \tau=0. \end{align*}

Here $y'(b)$ denotes the left derivative of $y$ at $b$.