Definition:Phase Space

Definition

Consider the flow:

$\map x t = \tuple {\map {x_1} t, \map {x_2} t, \ldots, \map {x_n} t}$

describing a solution to a differential equation in Euclidean $n$-space.

The phase space of this flow is the space of all vectors of the form:

$\tuple {\map {x_1} t, \map {\dot x_1} t, \map {x_2} t, \map {\dot x_1} t, \ldots, \map {x_n} t, \map {\dot x_n} t}$

in Euclidean $2 $-space.

Examples

Simple Harmonic Motion

Consider the differential equation:

$\ddot x + \omega^2 x 0$

A solution is:

$x = a \sin \omega t$

where $a$ and $\omega$ are positive constants.

Because:

$\dot x = a \omega \cos \omega t$

the phase space is the set of all points:

$\tuple {a \sin \omega t, a \omega \cos \omega t}$

which is the locus of an ellipse.

Also see

• Results about phase space can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): phase space
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): phase space