= Free solution of Hill equations
The planar <Hill equations> describe small displacements about a circular <Kepler orbit> in a frame rotating at angular frequency $n$. With $x$ radial and $y$ along the orbit, their unforced solution is
$$
x=A\cos nt+B\sin nt+C,\qquad
y=-2A\sin nt+2B\cos nt-\frac32nCt+D.
$$
Direct differentiation gives $\ddot x-2n\dot y-3n^2x=0$ and $\ddot y+2n\dot x=0$.
The $A,B$ terms describe <epicyclic motion> with tangential amplitude twice the radial amplitude. The constant radial offset $C$ causes an accumulating along-track displacement through <Keplerian shear>, while $D$ is an initial along-track offset. This separation makes the solution useful for comparing nearly neighbouring orbits and for identifying which motion requires additional encounter forcing.
Back to article page
