= Velocity-kick epicycle
{title2=$x/(\gamma a)=2\cos\theta(1-\cos nt)+\sin\theta\sin nt$}
A small planar <velocity> kick on a stellar <circular orbit> produces a local <epicyclic motion> in the source's <rotating reference frame>. With outward $x$, forward $y$, and initial displacements zero, <Hill equations> give $x/(\gamma a)=2\cos\theta(1-\cos nt)+\sin\theta\sin nt$ and $y/(\gamma a)=\cos\theta(4\sin nt-3nt)+2\sin\theta(\cos nt-1)$. Tangential kicks combine an epicycle with <Keplerian shear>; a radial kick gives a closed leading-order <epicyclic ellipse>.
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