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Stability, periodic orbits and KAM tori in the dynamics of the three fixed centers problem

Edward A. Turner, Francisco Crespo, Jhon Vidarte, Jersson Villafañe, Jorge Zapata

TL;DR

This work analyzes the motion of a test particle in the gravitational field of three fixed centers arranged at the vertices of an equilateral triangle, in a far-field regime. Using a two-stage Lie-Deprit normalization, the authors remove the mean anomaly and exploit the absence of the periapsis term to obtain a simplified Hamiltonian and two levels of reduction. They identify six relative equilibria corresponding to circular orbits in the reduced spaces, establishing parametric stability for the equatorial and polar families and instability for some polar cases. Employing Han–Li–Yi high-degeneracy KAM theory, they prove the persistence of abundant 3-tori around the stable near-circular periodic solutions in both the first and second reduced spaces, with the complement measure decaying as $O(\varepsilon^{\delta/4})$ for $0<\delta<1/5$, thereby illuminating the long-term stability and rich quasi-periodic structure of the system.

Abstract

We investigate the motion in space of an infinitesimal particle in the gravitational field generated by three primary bodies positioned at the vertices of a fixed equilateral triangle. We assume that the distances between the primaries are small compared to their separation from the particle. By applying a Lie-Deprit normalization, we simplify the Hamiltonian, relegating both the mean anomaly and the argument of periapisis to third-order terms or higher. After reducing out the symmetries associated with the Kepler flow and the central action of the angular momentum, we examine the relative equilibria in the first and second reduced spaces. We are able to identify the conditions for the existence of circular periodic orbits and KAM tori, thus providing insight into the system's long-term stability and dynamic structure.

Stability, periodic orbits and KAM tori in the dynamics of the three fixed centers problem

TL;DR

This work analyzes the motion of a test particle in the gravitational field of three fixed centers arranged at the vertices of an equilateral triangle, in a far-field regime. Using a two-stage Lie-Deprit normalization, the authors remove the mean anomaly and exploit the absence of the periapsis term to obtain a simplified Hamiltonian and two levels of reduction. They identify six relative equilibria corresponding to circular orbits in the reduced spaces, establishing parametric stability for the equatorial and polar families and instability for some polar cases. Employing Han–Li–Yi high-degeneracy KAM theory, they prove the persistence of abundant 3-tori around the stable near-circular periodic solutions in both the first and second reduced spaces, with the complement measure decaying as for , thereby illuminating the long-term stability and rich quasi-periodic structure of the system.

Abstract

We investigate the motion in space of an infinitesimal particle in the gravitational field generated by three primary bodies positioned at the vertices of a fixed equilateral triangle. We assume that the distances between the primaries are small compared to their separation from the particle. By applying a Lie-Deprit normalization, we simplify the Hamiltonian, relegating both the mean anomaly and the argument of periapisis to third-order terms or higher. After reducing out the symmetries associated with the Kepler flow and the central action of the angular momentum, we examine the relative equilibria in the first and second reduced spaces. We are able to identify the conditions for the existence of circular periodic orbits and KAM tori, thus providing insight into the system's long-term stability and dynamic structure.
Paper Structure (10 sections, 8 theorems, 72 equations, 2 figures)

This paper contains 10 sections, 8 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dynamics of a particle far from a triple-star system
  3. First reduced space
  4. Relative Equilibria and Stability
  5. KAM 3-Tori
  6. Second reduced space
  7. KAM Tori
  8. Reeb's Theorem and Parametrical Stable
  9. KAM Theory: Han Li and Yi's Theorem
  10. Complete Expression of $\mathcal{H}_2^0$

Key Result

Theorem 3.1

The relative equilibira $\mathbf E_i$ reconstruct into circular periodic orbits in the full system, with period $T=2\pi L^3+O(\varepsilon^3).$

Figures (2)

  • Figure 1: Simulation in the configuration space of the full system \ref{['eq:Ham3centros']} of the periodic orbits reconstructed from \ref{['eq:rev1eq']}. We consider $L=1$ and $\varepsilon=0.1$. $\mathbf{E}_{1,2}$ correspond with equatorial orbits illustrated in (a), while $\mathbf{E}_{3,4}$ and $\mathbf{E}_{5,6}$ reconstruct to the polar orbits showed in (b) and (c) respectively.
  • Figure 2: Plot of the function $f(z) = \Vert z(t_0) - z(t) \Vert$, with $z=(q,p)$.

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • Definition 1: parametrically stability
  • Theorem A.1: Krein-Gel'fand
  • Theorem A.2: Reeb
  • ...and 2 more