---
tag: CPEC2NW2KF
kind: concept
name: Phase space
msc: [34C99, 37C10, 70H05]
sources:
  - url: https://en.wikipedia.org/wiki/Phase_space
same_as:
  wikidata: Q62542
  wikipedia: Phase space
---

## Summary

The phase space of a system of differential equations is the set of all its possible states.
Each solution traces a curve in it, its [[trajectory|0BVT6VSB3S]], and the collection of these curves is the [[phase portrait|0T3GK0XCBC]].
Thinking in the phase space turns questions about formulas for solutions into questions about the geometry of curves.
See the [[definition|RY2RAH3802]].

## Explanations

### L1 human

A state is everything one needs to know now to predict the future.
For a moving particle this is its position together with its velocity, not the position alone.
So the phase space of a particle on a line is a plane, with position on one axis and velocity on the other.
For the [[harmonic oscillator|0YG0ZJPTWG]] $x'' = -x$, the motions are circles around the origin in this plane.
The [[phase plane|05VDGH129F]] and the [[phase line|081QSEAGEY]] are the phase spaces of dimension two and one.