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Computational symplectic topology and symmetric orbits in the restricted three-body problem

Chankyu Joung, Otto van Koert

TL;DR

This work addresses Birkhoff's disk-like global surface of section conjecture for the planar restricted three-body problem by integrating validated numerics with symplectic topology. It develops a computational framework around Levi-Civita and Moser regularizations, defining robust Conley-Zehnder index computations and action bounds for symmetric periodic orbits. The main results include the existence and non-degeneracy of a direct symmetric orbit and a retrograde orbit in a specified parameter regime, along with unknottedness and a self-linking number of $-1$, thereby linking to disk-like global surfaces of section via dynamical convexity. The approach demonstrates a practical path to proving dynamical convexity and informs potential extensions toward a full Birkhoff conjecture via covering arguments and index theory.

Abstract

In this paper we propose a computational approach to proving the Birkhoff conjecture on the restricted three-body problem, which asserts the existence of a disk-like global surface of section. Birkhoff had conjectured this surface of section as a tool to prove existence of a direct periodic orbit. Using techniques from validated numerics we prove the existence of an approximately circular direct orbit for a wide range of mass parameters and Jacobi energies. We also provide methods to rigorously compute the Conley-Zehnder index of periodic Hamiltonian orbits using computational tools, thus giving some initial steps for developing computational Floer homology and providing the means to prove the Birkhoff conjecture via symplectic topology. We apply this method to various symmetric orbits in the restricted three-body problem.

Computational symplectic topology and symmetric orbits in the restricted three-body problem

TL;DR

This work addresses Birkhoff's disk-like global surface of section conjecture for the planar restricted three-body problem by integrating validated numerics with symplectic topology. It develops a computational framework around Levi-Civita and Moser regularizations, defining robust Conley-Zehnder index computations and action bounds for symmetric periodic orbits. The main results include the existence and non-degeneracy of a direct symmetric orbit and a retrograde orbit in a specified parameter regime, along with unknottedness and a self-linking number of , thereby linking to disk-like global surfaces of section via dynamical convexity. The approach demonstrates a practical path to proving dynamical convexity and informs potential extensions toward a full Birkhoff conjecture via covering arguments and index theory.

Abstract

In this paper we propose a computational approach to proving the Birkhoff conjecture on the restricted three-body problem, which asserts the existence of a disk-like global surface of section. Birkhoff had conjectured this surface of section as a tool to prove existence of a direct periodic orbit. Using techniques from validated numerics we prove the existence of an approximately circular direct orbit for a wide range of mass parameters and Jacobi energies. We also provide methods to rigorously compute the Conley-Zehnder index of periodic Hamiltonian orbits using computational tools, thus giving some initial steps for developing computational Floer homology and providing the means to prove the Birkhoff conjecture via symplectic topology. We apply this method to various symmetric orbits in the restricted three-body problem.
Paper Structure (19 sections, 23 theorems, 86 equations, 5 figures)

This paper contains 19 sections, 23 theorems, 86 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hill's region
  4. Levi-Civita and Moser regularization
  5. Moser regularization
  6. Contact type and action
  7. Frames and local invariants of orbits
  8. Degree definition of Conley-Zehnder index
  9. Symmetries
  10. Symmetric periodic orbits
  11. Parametrizing the fixed point locus
  12. Crossing number
  13. Validated numerics
  14. Proofs
  15. Crossing theorem
  16. ...and 4 more sections

Key Result

Theorem 1.1

For all $(\mu,c)$ with $\mu \in[0.1,0.5]$ and Jacobi energy $c$ between and including the first critical energy and $2.1$, there is a symmetric, periodic orbit that is direct and crosses the $q_1$-axis exactly twice. This orbit is non-degenerate.

Figures (5)

  • Figure 1: Retrograde orbits and orbits that are not apparently retrograde for $\mu=0.9$
  • Figure 2: Plot of the boundaries of the Hill's regions for $\mu=0.1$ for various energy levels, with the Lagrange points indicated as dots. The paper is mostly concerned with orbits lying in the center region.
  • Figure 3: Schematic plot of the lift $\vartheta$ of the extension $\rho(\tilde{\psi}(t))$ to the cover ${\mathbb{R}} \to S^1=U(1)$ for two cases, along with their corresponding Conley-Zehnder indices.
  • Figure 4: Family of periodic orbits for $\mu=0.99$ which goes through a period doubling bifurcation.
  • Figure 5: Phase portrait of the Poincaré map with section $q_2=0$ in a neighborhood of the periodic orbit family of Theorem \ref{['thm:index']}. The phase portrait before (left) and after (right) the period doubling bifurcation show the transition from an elliptic to a hyperbolic orbit.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Hryniewicz
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • ...and 46 more