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  "body": "\n## Summary\n\nThe phase space of a system of differential equations is the set of all its possible states.\nEach solution traces a curve in it, its [[trajectory|0BVT6VSB3S]], and the collection of these curves is the [[phase portrait|0T3GK0XCBC]].\nThinking in the phase space turns questions about formulas for solutions into questions about the geometry of curves.\nSee the [[definition|RY2RAH3802]].\n\n## Explanations\n\n### L1 human\n\nA state is everything one needs to know now to predict the future.\nFor a moving particle this is its position together with its velocity, not the position alone.\nSo the phase space of a particle on a line is a plane, with position on one axis and velocity on the other.\nFor the [[harmonic oscillator|0YG0ZJPTWG]] $x'' = -x$, the motions are circles around the origin in this plane.\nThe [[phase plane|05VDGH129F]] and the [[phase line|081QSEAGEY]] are the phase spaces of dimension two and one.\n",
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