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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$, let $f : U \\to \\mathbb{R}^n$ be locally Lipschitz, and let $x^* \\in U$ be an [[equilibrium point|0G4X36AE0X]] of $x' = f(x)$.\nLet $W \\subseteq U$ be an open neighbourhood of $x^*$ and let $V : W \\to \\mathbb{R}$ be continuously differentiable.\nThe orbital derivative of $V$ is the function $\\dot{V}(x) = \\nabla V(x) \\cdot f(x)$ on $W$.\nThe function $V$ is a Lyapunov function for $x^*$ if $V(x^*) = 0$, $V(x) \u003e 0$ for each $x \\in W \\setminus \\{x^*\\}$, and $\\dot{V}(x) \\le 0$ for each $x \\in W$.\nIt is a strict Lyapunov function if also $\\dot{V}(x) \u003c 0$ for each $x \\in W \\setminus \\{x^*\\}$.\n\n## Remarks\n\nFor each solution $\\gamma$ in $W$, $\\frac{d}{dt} V(\\gamma(t)) = \\dot{V}(\\gamma(t))$. So $V$ does not increase along solutions.\nFor $x' = -x$ on $\\mathbb{R}$, the function $V(x) = x^2$ is a strict Lyapunov function for $0$, since $\\dot{V}(x) = -2x^2$.\nIn mechanics, the total energy is often a Lyapunov function. See [[Lyapunov's stability theorem|0J6A752DTZ]].\n\n## Explanations\n\n### L1 human\n\nA Lyapunov function is like a height above the equilibrium, which is the lowest point.\nIf the height never increases along the motion, the state cannot climb away from the equilibrium.\n",
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