Spherical isochrone model

The spherical isochrone model has potentialIts radial period depends only on orbital energy, while its radial action and frequencies are elementary functions of the integrals of motion.

For specific energy $E<0$ and specific angular momentum $L$,Consequently the Hamiltonian in actions is

The radial and azimuthal frequencies obeyThe ratio tends to one half in the harmonic core and to one in the Keplerian exterior.