Epicycle in a Rotating Frame

Consider a particle in circular orbit with angular frequency , and now jump to a frame which rotates with the particle -- so it remains stationary. Give the particle a radial perturbation (so its AM remains constant). Under the constraint of constant AM, the centripetal (centrifugal) force drops faster than the gravitational force. The particle enters a retrograde epicycle of angular frequency , as the gravitational and centrifugal forces alternate in dominance, and as the azimuthal velocity alternates between leading and lagging the guiding center.

In the case of a Keplerian potential, the epicycle is an ellipse with aspect ratio 2:1, with epicyclic frequency equal to the orbital frequency: = .