Isotropic harmonic oscillator flow
= Isotropic harmonic oscillator flow
{title2=$S_t=\begin{pmatrix}\cos\omega t\,I&(\sin\omega t/\omega)I\\-\omega\sin\omega t\,I&\cos\omega t\,I\end{pmatrix}$}
For $H_\omega=(|v|^2+\omega^2|x|^2)/2$, the displayed linear <Hamiltonian flow> solves $\dot X=V$, $\dot V=-\omega^2X$. Its inverse is $S_{-t}$, its determinant is one and it preserves $H_\omega$. The zero-frequency limit is the shear $(x,v)\mapsto(x+tv,v)$.