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ECH Constraints and Twist Dynamics in the Spatial Isosceles Three-Body Problem

Xijun Hu, Lei Liu, Yuwei Ou, Zhiwen Qiao, Pedro A. S. Salomão

Abstract

We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH classification of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. The result admits a dynamical interpretation via disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a non-trivial twist interval which encodes the relative winding of periodic orbits. For energies above the critical level, where the energy surface is non-compact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity.

ECH Constraints and Twist Dynamics in the Spatial Isosceles Three-Body Problem

Abstract

We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH classification of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. The result admits a dynamical interpretation via disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a non-trivial twist interval which encodes the relative winding of periodic orbits. For energies above the critical level, where the energy surface is non-compact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity.
Paper Structure (9 sections, 33 theorems, 206 equations, 3 figures)

This paper contains 9 sections, 33 theorems, 206 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['thm: main1']}
  3. Proof of Theorem \ref{['thm: main2']}
  4. Volume estimates
  5. Proof of Theorem \ref{['thm: main4']}
  6. Proof of Theorem \ref{['thm: main5']}
  7. Maslov-type index
  8. Relative winding number and linking number
  9. Proof of Lemma \ref{['lem: trace and determinant are positive']}

Key Result

Theorem 1.1

For every $(\beta,\mathfrak{e})\in (0,1) \times [0,1)$ denote the rotation number of the Euler orbit $\zeta_e$ by $\rho_{\beta,\mathfrak e}:=\rho_e$. Then for every $\beta\in (0,1)$ the function $\mathfrak{e} \mapsto \rho_{\beta,\mathfrak{e}}, \mathfrak{e} \in [0,1),$ is non-decreasing. Moreover,

Figures (3)

  • Figure 1.1: Singular foliation in the unbounded energy surface
  • Figure 2.1: The curves $\Gamma_j^1$ and $\Gamma_{j,\pm}$ in $[0,1] \times [0,1]$.
  • Figure 5.1: The Hamiltonian vector field of $H$ in $(x,y)$-coordinates.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 2.1: HLOSY2023, HOT2023
  • Theorem 2.2
  • ...and 50 more