Relative periodic solutions in spatial Kepler problem with symmetric perturbation

The spatial Kepler problem with a perturbation satisfying the rotational symmetry w.r.t. the $z$-axis and the reflection symmetry w.r.t. the $(x, y)$-plane, can be reduced to an Hamiltonian system with 2 degrees of freedom after fixing the angular momentum. For small enough perturbations, we show that for certain choices of energy and angular momentum, the corresponding energy surface is compact and diffeomorphic to $\mathbb{S}^3$, and on each compact energy surface there is a unique $z$-symmetric brake orbit, which forms a Hopf link with a planar relative periodic orbit. Moreover under some additional technical assumptions, by applying recent results from symplectic dynamics (\cite{CHHL23}) and Franks’ Theorem, we prove there are infinitely many relative periodic orbits on each compact energy surface. These results can be applied to the motion of a satellite around a uniformly mass-distributed ellipsoid and the $n$-pyramidal problem, where one point mass moves along the $z$-axis and $n$ other equal point masses form a regular $n$-gon perpendicular to the $z$-axis.

💡 Research Summary

The paper investigates a spatial Kepler problem perturbed by a smooth potential that is invariant under rotations about the z‑axis and under reflection through the (x,y)‑plane. Such a perturbation naturally arises when modelling the gravitational field of a uniformly mass‑distributed ellipsoid with small eccentricity, or in the so‑called n‑pyramidal configuration where one mass moves on the z‑axis while n equal masses form a regular n‑gon in a plane perpendicular to that axis.

By fixing the angular momentum ω about the z‑axis, the three‑degree‑of‑freedom Hamiltonian system reduces to a two‑degree‑of‑freedom Hamiltonian

 H_{ω,ε}(p_r,p_z,r,z)=½(p_r²+p_z²)+ω²/(2r²)−1/√(r²+z²)+ε f(r,z,ε),

where ε∈