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Relative periodic solutions in spatial Kepler problem with symmetric perturbation

Xijun Hu, Zhiwen Qiao, Guowei Yu

TL;DR

This work analyzes a spatial Kepler problem perturbed by a symmetric potential that preserves rotation about the $z$-axis and reflection across the $xy$-plane. By fixing the $z$-axis angular momentum, the system reduces to a 2-DoF Hamiltonian whose energy surfaces are compact and diffeomorphic to ${\mathbb S}^3$ for small perturbations, enabling the use of contact geometry and Reeb dynamics to locate global surfaces of section and to connect action, volume, and rotation data. The authors prove the existence of a unique $z$-symmetric brake orbit forming a Hopf link with a planar relative periodic orbit on each energy surface; under additional hypotheses, they invoke recent results and Franks' theorem to obtain infinitely many relative periodic orbits. The framework is then specialized to the ellipsoid problem and the $n$-pyramidal problem, showing infinite families of relative periodic orbits for small eccentricities or mass ratios, respectively, and providing explicit criteria via the Maslov-type index and related invariants. Overall, the paper advances a symplectic-geometric route to multiplicity results for relative periodic motions in symmetric celestial-mechanical models.

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.

Relative periodic solutions in spatial Kepler problem with symmetric perturbation

TL;DR

This work analyzes a spatial Kepler problem perturbed by a symmetric potential that preserves rotation about the -axis and reflection across the -plane. By fixing the -axis angular momentum, the system reduces to a 2-DoF Hamiltonian whose energy surfaces are compact and diffeomorphic to for small perturbations, enabling the use of contact geometry and Reeb dynamics to locate global surfaces of section and to connect action, volume, and rotation data. The authors prove the existence of a unique -symmetric brake orbit forming a Hopf link with a planar relative periodic orbit on each energy surface; under additional hypotheses, they invoke recent results and Franks' theorem to obtain infinitely many relative periodic orbits. The framework is then specialized to the ellipsoid problem and the -pyramidal problem, showing infinite families of relative periodic orbits for small eccentricities or mass ratios, respectively, and providing explicit criteria via the Maslov-type index and related invariants. Overall, the paper advances a symplectic-geometric route to multiplicity results for relative periodic motions in symmetric celestial-mechanical models.

Abstract

The spatial Kepler problem with a perturbation satisfying the rotational symmetry w.r.t. the -axis and the reflection symmetry w.r.t. the -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 , and on each compact energy surface there is a unique -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 -pyramidal problem, where one point mass moves along the -axis and other equal point masses form a regular -gon perpendicular to the -axis.

Paper Structure

This paper contains 9 sections, 26 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. The Kepler Problem
  3. The Kepler Problem with a symmetric perturbation
  4. Estimate the contact volume and action value
  5. Seifert rotation number of $\xi_{\varepsilon}$
  6. The existence of infinitely many periodic orbits and a unique brake $z$-symmetric orbit which forms a Hopf link with $\xi_{\varepsilon}$.
  7. The ellipsoid problem
  8. The n-pyramidal problem
  9. Appendix. The Maslov-type index for symplectic paths

Key Result

Theorem 1.1

The ellipsoid problem admits infinitely many relative periodic solutions within the compact energy surface $\mathcal{M}(h,\varpi,\mathfrak{e})$ for sufficiently small eccentricity $\mathfrak{e}$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Proposition 1.1
  • Remark 1.1
  • Theorem 1.2
  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.3
  • Remark 1.4
  • ...and 50 more