Editing Oscillation (section)

== Two-dimensional oscillators ==
In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an [[Isotropy|isotropic]] oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions.
<math display="block">\vec{F} = -k\vec{r}</math>

This produces a similar solution, but now there is a different equation for every direction.
<math display="block">\begin{align}
x(t) &= A_x \cos(\omega t - \delta _x), \\
y(t) &= A_y \cos(\omega t - \delta_y), \\
& \;\, \vdots
\end{align}</math>

=== Anisotropic oscillators ===
With [[Anisotropy|anisotropic]] oscillators, different directions have different constants of restoring forces. The solution is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is [[Quasiperiodic function|quasiperiodic]]. This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat.<ref name=":0">{{Cite book |last=Taylor |first=John R. |url=https://www.worldcat.org/oclc/55729992 |title=Classical mechanics |date=2005 |isbn=1-891389-22-X |location=Mill Valley, California |oclc=55729992}}</ref>