A number of people (including Newton himself) have noticed that Newton's elliptical solution for orbits in an inverse square central force field
Julian Schwinger used to claim that all movements in force fields were equivalent to harmonic oscillators. The proof was similar to the traditional use of the Hamilton-Jacobi equation to reinterpret complicated motion in a simple coordinate system as simple (eg, force-free) motion in a complicated coordinate system. A nice theory but not overly helpful in practical situations. The exchange between the harmonic oscillator and Kepler's ellipses has an extensive history, some of which is in those cited references. The quantum mechanical analysis of the "accidental" degeneracy of the hydrogen atom depends on this interrelation, leading to O(4) or SU(2,2) as symmetry groups. -hvm