Hill equations

The unforced planar equation of motion in a rotating frame near a circular Kepler orbit has components $\ddot{x}-2n\dot{y}-3n^{2}x=0$ and $\ddot{y}+2n\dot{x}=0$. They describe epicyclic motion and Keplerian shear; mutual gravity adds close-encounter forcing.

The oscillatory integration constants determine the orbital eccentricity vector, the constant radial offset determines the semi-major axis, and the constant along-track offset fixes the mean longitude. The radial and tangential epicycle amplitudes have ratio $1:2$.