Isotropic harmonic oscillator flow ($S_t=\left(\begin{array}{cc}\cos \omega tI& (\sin \omega t/\omega )I\\ -\omega \sin \omega tI& \cos \omega tI\end{array}\right)$)

For $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$, the displayed linear Hamiltonian flow solves $\dot{X}=V$, $\dot{V}=-{\omega }^{2}X$. Its inverse is $S_{-t}$, its determinant is one and it preserves $H_{\omega }$. The zero-frequency limit is the shear $(x,v)\mapsto (x+tv,v)$.