= Astroid motion of a harmonic oscillator in a rotating frame
{title2=$(x',y')=(\cos^3\omega t,\sin^3\omega t)$}
For an isotropic <harmonic oscillator> with inertial frequency $2\omega$, viewed in a frame rotating at $\omega$, initial rotating coordinates $(1,0)$ and zero relative <velocity> produce the displayed <astroid> trajectory. The complex coordinate satisfies $\ddot\zeta+2i\omega\dot\zeta+3\omega^2\zeta=0$, with solution $\zeta=\tfrac34e^{i\omega t}+\tfrac14e^{-3i\omega t}$. The relative <speed> is $(3|\omega|/2)|\sin2\omega t|$, attaining its first maximum at $\pi/(4|\omega|)$ for nonzero $\omega$.
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