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  "body": "\n## Definition\n\nLet $U$ be an open subset of $\\mathbb{R}^n$ and let $f : U \\to \\mathbb{R}^n$ be locally Lipschitz. Let $\\varphi$ be the [[flow|02903SSA48]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$.\nLet $x_0 \\in U$, and let $I(x_0)$ be the maximal interval of existence of the solution with $x(0) = x_0$.\nThe trajectory, or orbit, of $x_0$ is the set\n$$\\mathcal{O}(x_0) = \\{\\varphi_t(x_0) : t \\in I(x_0)\\} \\subseteq U.$$\nThe positive half-trajectory is $\\mathcal{O}^+(x_0) = \\{\\varphi_t(x_0) : t \\in I(x_0),\\ t \\ge 0\\}$, and the negative half-trajectory uses $t \\le 0$.\n\n## Remarks\n\nTwo trajectories are either equal or disjoint. So the trajectories form a partition of $U$.\nA trajectory is a single point exactly when $x_0$ is an [[equilibrium point|0G4X36AE0X]]. It is a closed curve when the solution is [[periodic|01DSNP0B4R]] and not constant.\nSome books use the word trajectory for the solution curve $t \\mapsto \\varphi_t(x_0)$ itself, and the word orbit for its image.\n\n## Explanations\n\n### L1 human\n\nThe trajectory is the path that the state follows in the phase space.\nIt forgets the time: it records where the state goes, not when.\n",
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