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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be an open set and let $f : U \\to \\R^n$ be a continuous map.\nThe *phase space* of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ is the set $U$.\nIts points are called states, and the curve $t \\mapsto x(t)$ of a solution in $U$ is its [[trajectory|0BVT6VSB3S]].\nFor an [[ordinary differential equation|0P97V20RW7]] $y^{(n)} = g(y, y', \\dots, y^{(n-1)})$ of order $n$, the phase space is the phase space of the equivalent first-order system for the state $(y, y', \\dots, y^{(n-1)})$.\n\n## Remarks\n\nExample: the equation $x'' = -x$ becomes the system $x' = v$, $v' = -x$ on $U = \\R^2$.\nIts phase space is the plane with coordinates $(x, v)$.\nThe solution $x = \\cos t$ has the trajectory $(\\cos t, -\\sin t)$, which runs on the unit circle, since $\\cos^2 t + \\sin^2 t = 1$.\nNon-example: for a pendulum, the set of angles alone is not the phase space.\nThe angle at one time does not determine the motion; the angular velocity is also needed.\nFor a non-autonomous system $x' = f(t, x)$ on an open set $D \\subseteq \\R \\times \\R^n$, the set $D$ is called the extended phase space.\nFor a [[Hamiltonian system|0C2AKPDCD9]] with $n$ degrees of freedom, the phase space has the $2n$ coordinates $(q_1, \\dots, q_n, p_1, \\dots, p_n)$ of positions and momenta.\n",
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