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.