Normal forms for the epicyclic approximations of the perturbed Kepler problem

Abstract

We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the tidal expansion of the potential, the equations of motion take the form of perturbed harmonic oscillators in a rotating frame. In the unperturbed, purely Keplerian case, the post-epicyclic solutions produced with the normal form coincide with those obtained from the expansion of the solution of the Kepler equation. In all cases where the perturbed problem can be cast in autonomous form, the solution is easily obtained as a perturbation series. The generalization to the spatial problem and/or the non-autonomous case is straightforward.

Halo orbits around the collinear points of the restricted three-body problem

“…However, the finite value of the threshold for µ → 0 must still be considered a consistent prediction in the light of the singularity of the perturbation problem. A hint to the reliability of this prediction comes from the observation that the rotation number (23) around the Lyapunov planar orbit tends to zero (see also [28], Section 4.5) and the time-scale of instability diverges in time with a rate exponentially small in µ. These statements are easily verified by using (22) and observing that every term appearing in the argument of the square root vanishes with µ.…”

“…, perturbed by terms with coefficients c n = (−1) n +C n µ with the C n numbers of order one, all terms of degree zero in µ disappear from the normal form [23].…”

“…y ∼ κ with the C n numbers of order one, all terms of degree zero in µ disappear from the normal form [23].…”

Bifurcation Sequences in the Symmetric 1:1 Hamiltonian Resonance

“…The first two of them provide the frequencies of the epicyclic motions [Pucacco, 2012]. Recalling the time rescaling in (161), the radial and vertical harmonic frequencies are respectively…”