Let $a,b \in \mathbb{R}$ with $a < b$, let $n \in \mathbb{N}$, and let $L \in C^1([a,b]\times \mathbb{R}^n \times \mathbb{R}^n;\mathbb{R})$. Let $y \in C^1([a,b];\mathbb{R}^n)$, and assume that $y$ satisfies the weak Euler-Lagrange equation: for every variation $\varphi \in C_c^1((a,b);\mathbb{R}^n)$,
where $\partial_y L$ denotes the derivative of $L$ with respect to its second argument and $\partial_v L$ denotes the derivative of $L$ with respect to its third argument. If the map
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\begin{align*}
x \mapsto \partial_v L(x,y(x),y'(x))
\end{align*}
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belongs to $C^1((a,b);\mathbb{R}^n)$, then $y$ satisfies the classical Euler-Lagrange equation