Relative periodic solutions in spatial Kepler problem with symmetric perturbation

TL;DR

This paper studies the existence and multiplicity of relative periodic solutions in a symmetric spatial Kepler problem, revealing infinitely many such orbits under certain conditions using symplectic dynamics and topological methods.

Contribution

It demonstrates the existence of unique symmetric brake orbits and infinitely many relative periodic orbits in a perturbed Kepler system with symmetry, extending previous dynamical results.

Findings

Existence of a unique $z$-symmetric brake orbit on each energy surface.

Presence of a Hopf link between brake and planar relative periodic orbits.

Infinitely many relative periodic orbits under additional assumptions.

Abstract

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.