Phase Space/Examples/Simple Harmonic Motion

Example of Phase Space

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.

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