---
tag: RY2RAH3802
kind: definition
name: Phase space
msc: [34C99, 37C10, 70H05]
defines: CPEC2NW2KF
uses: [07BNRFQ0PJ, 0BVT6VSB3S, 0P97V20RW7]
sources:
  - url: https://en.wikipedia.org/wiki/Phase_space
---

## Definition

Let $U \subseteq \R^n$ be an open set and let $f : U \to \R^n$ be a continuous map.
The *phase space* of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ is the set $U$.
Its points are called states, and the curve $t \mapsto x(t)$ of a solution in $U$ is its [[trajectory|0BVT6VSB3S]].
For 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 $(y, y', \dots, y^{(n-1)})$.

## Remarks

Example: the equation $x'' = -x$ becomes the system $x' = v$, $v' = -x$ on $U = \R^2$.
Its phase space is the plane with coordinates $(x, v)$.
The solution $x = \cos t$ has the trajectory $(\cos t, -\sin t)$, which runs on the unit circle, since $\cos^2 t + \sin^2 t = 1$.
Non-example: for a pendulum, the set of angles alone is not the phase space.
The angle at one time does not determine the motion; the angular velocity is also needed.
For 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.
For 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.