= Isotropic harmonic oscillator
{title2=$V(\mathbf r)=m\Omega^2|\mathbf r|^2/2$}
The classical <isotropic harmonic oscillator> has the same <angular frequency> $\Omega>0$ in every Cartesian direction. Its <equation of motion> is $\ddot{\mathbf r}=-\Omega^2\mathbf r$, with solution $\mathbf r=\mathbf A\cos\Omega t+\mathbf B\sin\Omega t$. The trajectory lies in the span of $\mathbf A,\mathbf B$ and is an <ellipse> centred on the force centre, degenerating to a line for dependent vectors. For nonzero <angular momentum>, $r^2$ oscillates at $2\Omega$ unless the trajectory is exactly circular. This distinguishes radial and Cartesian <frequencies>.
Back to article page
