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Dynamics of an isosceles problem generated by a perturbation of Euler's collinear solution

Karine Santos

TL;DR

This work analyzes the linear and parametric stability of a spatial isosceles three-body problem arising from a perturbation of Euler's collinear solution. By transforming to rotating-pulsating coordinates and performing a symplectic reduction anchored by a first integral, the authors obtain a two-degree-of-freedom, time-periodic Hamiltonian with a single equilibrium at $\gamma=1$, and prove its linear stability for $\epsilon=0$ via Dirichlet’s theorem. They then investigate parametric stability under small eccentricity $\epsilon$ and mass ratio $\mu$ using the Krein-Gelfand-Lidskii theorem to identify resonances, and apply Deprit–Hori normalization to construct an autonomous Hamiltonian $K$ up to fourth order, deriving boundary curves in the $(\epsilon,\mu)$ plane from $p(\lambda)=\lambda^4+a\lambda^2+b$. Although several double-resonance scenarios are worked out and explicit curves are presented, the necessary condition $a>0$ cannot be satisfied at the computed order with $b=0$, leaving instability boundaries inconclusive at this level. The results illuminate how resonances shape the parametric stability landscape of isosceles perturbations of Euler's configuration.

Abstract

This paper presents a study of the isosceles problem resulting by a perturbation of Euler's collinear solution under Newtonian gravitational attraction of three bodies in space. After the Hamiltonian was obtained, a circumference of relative equilibria points was found. The original system was subsequently reduced to another system with two degrees of freedom, periodic in the time, where there is now a single point of equilibrium. Linear and parametric stability were discussed in this simplified model of the three-body problem.

Dynamics of an isosceles problem generated by a perturbation of Euler's collinear solution

TL;DR

This work analyzes the linear and parametric stability of a spatial isosceles three-body problem arising from a perturbation of Euler's collinear solution. By transforming to rotating-pulsating coordinates and performing a symplectic reduction anchored by a first integral, the authors obtain a two-degree-of-freedom, time-periodic Hamiltonian with a single equilibrium at , and prove its linear stability for via Dirichlet’s theorem. They then investigate parametric stability under small eccentricity and mass ratio using the Krein-Gelfand-Lidskii theorem to identify resonances, and apply Deprit–Hori normalization to construct an autonomous Hamiltonian up to fourth order, deriving boundary curves in the plane from . Although several double-resonance scenarios are worked out and explicit curves are presented, the necessary condition cannot be satisfied at the computed order with , leaving instability boundaries inconclusive at this level. The results illuminate how resonances shape the parametric stability landscape of isosceles perturbations of Euler's configuration.

Abstract

This paper presents a study of the isosceles problem resulting by a perturbation of Euler's collinear solution under Newtonian gravitational attraction of three bodies in space. After the Hamiltonian was obtained, a circumference of relative equilibria points was found. The original system was subsequently reduced to another system with two degrees of freedom, periodic in the time, where there is now a single point of equilibrium. Linear and parametric stability were discussed in this simplified model of the three-body problem.
Paper Structure (10 sections, 6 theorems, 52 equations, 3 figures)

This paper contains 10 sections, 6 theorems, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Hamiltonian in rotating-pulsating coordinates
  4. Relative Equilibria
  5. Characteristic equation of Hamiltonian system
  6. Null eigenvalue elimination and normal form of the reduced Hamiltonian
  7. Parametric stability
  8. Boundary curves of the stability/instability regions
  9. Conclusions
  10. Acknowledgments

Key Result

Proposition 2.1

There are vector functions $\overline{v}=\overline{v}(t)$ and $\overline{w}=\overline{w}(t)$ such that the vectors $r_1,\;r_2$ and $r_3$ at r1r2r3 and r3 are three-body problem solutions.

Figures (3)

  • Figure 1: Collinear Euler's solution.
  • Figure 2: Perturbation of Euler's collinear solution.
  • Figure 3: Geometric interpretation of equilibria circumference.

Theorems & Definitions (12)

  • Proposition 2.1
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Proposition 6.1
  • proof
  • Proposition 6.2
  • Proposition 6.3
  • proof
  • ...and 2 more