Symmetry associated to a part of a separable Hamiltonian

Question

The harmonic oscillator in 3D is: $$H=\frac{p_x^2+p_y^2+p_z^2}{2m}+ \frac{k}{2} (x^2+y^2+z^2) = H_x + H_y + H_z,$$

where $H_x$, $H_y$ and $H_z$ are all constants of motion (alongside $\vec{L}$). Time translation invariance implies the conservation of $H$. What is the symmetry associated to the conservation of $H_x$, $H_y$ and $H_z$? I guess it's linked to time invariance as well?

Answer 1

Well, quite generally for Hamiltonian systems, the infinitesimal symmetry behind a constant of motion (COM) is the Hamiltonian vector field generated by the COM itself, cf. this Phys.SE post.