Let $a,b \in \mathbb{R}$ with $a < b$, let $n \in \mathbb{N}$, let $\mathcal A \subset C^1([a,b];\mathbb{R}^n)$, and let $J: \mathcal A \to \mathbb{R}$ be a functional. Equip $C^1([a,b];\mathbb{R}^n)$ with the norm
Suppose $y \in \mathcal A$ is a local minimiser or a local maximiser of $J$ with respect to this norm. Let $h \in C^1([a,b];\mathbb{R}^n)$ be an admissible variation at $y$, meaning that there exists $\eta > 0$ such that $y + \varepsilon h \in \mathcal A$ for every $\varepsilon \in (-\eta,\eta)$. If the map