Let $a<b$, let $n\in\mathbb N$, and let $U\subset [a,b]\times\mathbb R^n\times\mathbb R^n$ be relatively open. Let $L\in C^2(U;\mathbb R)$, and write points of $U$ as $(x,y,v)$ with $x\in [a,b]$, $y\in\mathbb R^n$, and $v\in\mathbb R^n$. Assume the strengthened Legendre condition holds on $U$: for every $(x,y,v)\in U$, the matrix
\begin{align*}
W:=\{(x,y,p)\in [a,b]\times\mathbb R^n\times\mathbb R^n:\text{ there exists }v\in\mathbb R^n\text{ with }(x,y,v)\in U\text{ and }p=\partial_vL(x,y,v)\}.
\end{align*}
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Assume that the Legendre map in the velocity variable has a unique $C^1$ inverse on the relatively [open set](/page/Open%20Set) $W$, denoted
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\begin{align*}
v:W\to\mathbb R^n,
\end{align*}
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so that for every $(x,y,p)\in W$ one has $(x,y,v(x,y,p))\in U$ and