Quantity conserved for the 3D spherically symmetric harmonic potential $V(r)=\alpha r^2$

Question

I know that in the case of the Kepler problem there is a quantity (other than energy, momentum,...) conserved which is the Runge-Lenz vector.

Is there also an "exotic" quantity conserved for a 2-Body system with a potential: $V(r)=\alpha r^2$? I'm asking this because the radial harmonic oscillator and the inverse square force are the only ones that have a bounded orbit.

If the answer is positive; is there a symmetry associated to the conservation of this quantity?

Does this answer your question? Is there an analog to the Runge-Lenz vector for a harmonic potential? — Void, Jan 10 '20 at 13:14

Qmechanic · Accepted Answer · 2020-01-10T15:03:52.370

The 3D spherically symmetric harmonic oscillator $$ H~=~\frac{p_x^2+p_y^2+p_z^2}{2m}+ \alpha (x^2+y^2+z^2) ~=~ H_x + H_y + H_z $$ is a separable, Liouville integrable, and in fact a maximally superintegrable system with additionally integrals of motion $H_x$, $H_y$, $H_z$, $L_x$, $L_y$ and $L_z$, i.e. nothing exotic like the Laplace-Runge-Lenz vector. (A 3D system can maximally have 5 independent integrals of motion, cf. e.g. this Phys.SE post.)

Quantity conserved for the 3D spherically symmetric harmonic potential $V(r)=\alpha r^2$

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