[proofplan] We prove Lyapunov stability by constructing, inside any prescribed neighbourhood of $x^*$, a smaller energy sublevel set that traps all nearby trajectories. The strict local minimum makes the energy on a small sphere around $x^*$ strictly larger than $H(x^*)$; compactness of the sphere turns this into a positive energy gap. Initial conditions chosen with energy below that gap cannot reach the sphere, because $H$ is conserved along solutions. Therefore the trajectory remains inside the chosen ball for all forward time. [/proofplan] [step:Choose a small sphere on which the energy is uniformly above $H(x^*)$] For every $a \in \mathbb{R}^n$ and every $s > 0$, define the open ball $B(a,s) := \{x \in \mathbb{R}^n : |x-a| < s\}$ and the closed ball $\overline{B}(a,s) := \{x \in \mathbb{R}^n : |x-a| \leq s\}$. Let $\varepsilon > 0$ be given. Since $U \subset \mathbb{R}^n$ is open and $x^* \in U$, there exists $r_U > 0$ such that $\overline{B}(x^*, r_U) \subset U$. Since $H$ has a strict local minimum at $x^*$, there exists $r_H > 0$ such that \begin{align*} 0 < |x - x^*| < r_H \implies H(x) > H(x^*). \end{align*} Let $\rho > 0$ be the forward-existence radius from the hypotheses. Choose \begin{align*} 0 < r < \min\{\varepsilon, r_U, r_H, \rho\}. \end{align*} Define the boundary sphere \begin{align*} S_r := \{x \in \mathbb{R}^n : |x - x^*| = r\}. \end{align*} The set $S_r$ is compact in $\mathbb{R}^n$. Define the [continuous function](/page/Continuous%20Function) $G: S_r \to \mathbb{R}$ by $G(x) := H(x) - H(x^*)$ for every $x \in S_r$. Since $H$ is continuous, $G$ is continuous. Since $r < r_H$, every $x \in S_r$ satisfies $H(x) > H(x^*)$. Hence $G$ is positive on $S_r$. The image $G(S_r)$ is compact in $\mathbb{R}$ because $S_r$ is compact and $G$ is continuous. Since $G(S_r) \subset (0,\infty)$, its least element is positive; define \begin{align*} m := \min_{x \in S_r} \bigl(H(x) - H(x^*)\bigr). \end{align*} Then $m > 0$. [guided] For every $a \in \mathbb{R}^n$ and every $s > 0$, define the open ball $B(a,s) := \{x \in \mathbb{R}^n : |x-a| < s\}$ and the closed ball $\overline{B}(a,s) := \{x \in \mathbb{R}^n : |x-a| \leq s\}$. Fix $\varepsilon > 0$. Our goal is to keep the trajectory inside $B(x^*,\varepsilon)$ forever. We first choose a smaller ball whose boundary has strictly larger energy than the equilibrium. Because $U$ is open and $x^* \in U$, there is a radius $r_U > 0$ with \begin{align*} \overline{B}(x^*, r_U) \subset U. \end{align*} Because $H$ has a strict local minimum at $x^*$, there is a radius $r_H > 0$ such that every point near $x^*$ but different from $x^*$ has strictly larger energy: \begin{align*} 0 < |x - x^*| < r_H \implies H(x) > H(x^*). \end{align*} Finally, the theorem assumes that there is a radius $\rho > 0$ such that all solutions starting in $B(x^*,\rho)$ exist for every forward time $t \geq 0$. We choose \begin{align*} 0 < r < \min\{\varepsilon, r_U, r_H, \rho\}. \end{align*} This single choice ensures four things at once: the ball is inside the desired $\varepsilon$-neighbourhood, the closed ball lies in the domain $U$, the strict-minimum inequality holds on the punctured ball, and initial data in $B(x^*,r)$ fall inside the forward-existence neighbourhood. Now define the sphere \begin{align*} S_r := \{x \in \mathbb{R}^n : |x - x^*| = r\}. \end{align*} Define the function $G: S_r \to \mathbb{R}$ by $G(x) := H(x) - H(x^*)$ for every $x \in S_r$. This function is continuous because $H$ is continuous. The sphere $S_r$ is compact in finite-dimensional Euclidean space. Since $r < r_H$, every point $x \in S_r$ satisfies $0 < |x-x^*| < r_H$, so \begin{align*} G(x) = H(x) - H(x^*) > 0. \end{align*} The compactness argument is as follows. Since $S_r$ is compact and $G$ is continuous, the image $G(S_r)$ is compact in $\mathbb{R}$. A nonempty compact subset of $\mathbb{R}$ has a least element, and because every value of $G$ is strictly positive, that least element is also strictly positive. Therefore the number \begin{align*} m := \min_{x \in S_r} \bigl(H(x) - H(x^*)\bigr) \end{align*} is well-defined and satisfies $m > 0$. This positive gap is the energy barrier that the trajectory will be unable to cross. [/guided] [/step] [step:Choose initial data with energy below the boundary energy barrier] Since $H$ is continuous at $x^*$ and $m > 0$, there exists $\delta_0 > 0$ such that \begin{align*} |x_0 - x^*| < \delta_0 \implies |H(x_0) - H(x^*)| < m. \end{align*} Set \begin{align*} \delta := \min\{\delta_0, r\}. \end{align*} If $|x_0 - x^*| < \delta$, then $x_0 \in B(x^*,\rho)$ because $\delta \leq r < \rho$, so every solution starting at $x_0$ exists for all $t \geq 0$. Moreover, \begin{align*} H(x_0) < H(x^*) + m. \end{align*} [/step] [step:Use conservation of energy to rule out first contact with the sphere] Let $x: [0,\infty) \to U$ be a solution of $\dot{x}=X(x)$ with $x(0)=x_0$ and $|x_0-x^*|<\delta$. Since $H$ is conserved along solutions, \begin{align*} H(x(t)) = H(x_0) \end{align*} for every $t \geq 0$. Hence \begin{align*} H(x(t)) < H(x^*) + m \end{align*} for every $t \geq 0$. We claim that $|x(t)-x^*|<r$ for every $t \geq 0$. Suppose not. Since $|x(0)-x^*|<\delta \leq r$ and the map $t \mapsto |x(t)-x^*|$ is continuous, there exists $t_1 \geq 0$ such that \begin{align*} |x(t_1)-x^*| = r. \end{align*} Thus $x(t_1) \in S_r$, so the definition of $m$ gives \begin{align*} H(x(t_1)) - H(x^*) \geq m. \end{align*} Equivalently, \begin{align*} H(x(t_1)) \geq H(x^*) + m. \end{align*} This contradicts the conserved-energy inequality $H(x(t_1)) < H(x^*) + m$. Therefore $|x(t)-x^*|<r$ for all $t \geq 0$. [/step] [step:Conclude Lyapunov stability from the trapped ball] Since $r < \varepsilon$, the estimate from the previous step implies \begin{align*} |x(t)-x^*| < \varepsilon \end{align*} for every $t \geq 0$. Thus for the given $\varepsilon > 0$ we have found $\delta > 0$ such that every solution starting within $\delta$ of $x^*$ remains within $\varepsilon$ of $x^*$ for all forward time. Since the theorem assumes $X(x^*)=0$, the point $x^*$ is an equilibrium, and this is precisely Lyapunov stability of the equilibrium $x^*$. [/step]