The laws of planetary motion, derived from those of a harmonic   oscillator (following Arnold)
  Kepler's laws of planetary motion are deduced from those of a harmonic oscillator following Arnold. Conversely, the circular orbits through the Earth's center suggested by Galilei are consistent with an $r^{-5}$ potential as found before by Newton. Both the Kepler/oscillator correspondance and circular orbits are examples of dual potentials. 
P. A. Horvathy and P.-M. Zhang
physics.class-ph gr-qc math-ph math.MP