Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory

Abstract: It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter $\bb=27(m_1m_2+m_2m_3+m_3m_1)/(m_1+m_2+m_3)^2\in [0,9]$ and the eccentricity $e\in [0,1)$. We are not aware of any existing analytical method which relates the linear stability of these solutions to the two parameters directly in the full rectangle $[0,9]\times [0,1)$, besides perturbation methods for $e>0$ small enough, blow-up techniques for $e$ sufficiently close to 1, and numerical studies. In this paper, we introduce a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full $(\bb,e)$ range $[0,9]\times [0,1)$ via the $\om$-index theory of symplectic paths for $\om$ belonging to the unit circle of the complex plane, and the theory of linear operators. After establishing the $\om$-index decreasing property of the solutions in $\bb$ for fixed $e\in [0,1)$, we prove the existence of three curves located from left to right in the rectangle $[0,9]\times [0,1)$, among which two are -1 degeneracy curves and the third one is the right envelop curve of the $\om$-degeneracy curves for $\om\not=1$, and show that the linear stability pattern of such elliptic Lagrangian solutions changes if and only if the parameter $(\bb,e)$ passes through each of these three curves. Interesting symmetries of these curves are also observed. The singular case when the eccentricity $e$ approaches to 1 is also analyzed in details concerning the linear stability.

• The paper presents a novel analytic method using Maslov-type ω-index theory to rigorously determine stability boundaries in the planar three-body problem.
• It classifies stability regions by analyzing critical mass and eccentricity parameters, identifying transitions via two -1 degeneracy curves and one Krein collision curve.
• The results establish a comprehensive framework for predicting bifurcations and linear stability, with implications for broader applications in celestial mechanics.

Linear Stability of Elliptic Lagrangian Solutions of the Planar Three-Body Problem via Index Theory

Introduction

The paper "Linear stability of elliptic Lagrangian solutions of the planar three-body problem via index theory" (1206.6162) presents a rigorous analytical treatment of the linear stability of elliptic Lagrangian (equilateral triangle homographic) solutions in the planar Newtonian three-body problem. The analysis leverages $\omega$-Maslov-type index theory for symplectic paths, providing a direct connection between linear stability and the primary motion parameters: the mass parameter $\beta$ and the orbital eccentricity $e$. This constitutes a significant advancement relative to existing results that relied heavily on perturbative, blow-up, or numerical methods for restricted subdomains of the $(\beta,e)$ parameter space.

Problem Context and Methodological Framework

Lagrange's 1772 construction of homographic solutions establishes that, in the planar three-body problem, the three bodies may follow the vertices of a rotating equilateral triangle, each following a Keplerian elliptic orbit about the barycenter. When $e=0$, this reduces to the classical equilateral triangle rotating (relative equilibrium); for $e>0$, the triangle distorts dynamically as the three orbits become elliptic.

Linear stability analysis of these orbits is pivotal for understanding the persistence of such structures under small perturbations. The parameterization is via the "mass parameter" $$\beta=27(m_1m_2 + m_2m_3 + m_3m_1)/(m_1+m_2+m_3)^2 \in [0,9]$$, a homogeneous, symmetric combination of the masses, and eccentricity $e\in[0,1)$.

The novelty here is the derivation of analytic results for the full range $(\beta,e) \in [0,9]\times[0,1)$ using index theory. Previous approaches either worked for small $e$ (perturbative), $e$ near $1$ (blow-up), or numerically; no analytic framework provided detailed stability boundaries throughout the whole parameter rectangle.

The methodology is based on the Maslov-type $\omega$-index for symplectic paths, specifically symplectic monodromy matrices arising from linearization about the solution. The theory connects the evolution of spectral properties (multipliers on the unit circle $\mathbb{U}$ of the complex plane) to the passage through critical curves in $(\beta,e)$.

Main Results

Rigorous Determination of Stability Boundaries

A key component is the identification and analysis of three distinct critical curves in $(\beta,e)$ space:

1. Two $-1$-degeneracy curves ($\Gamma_s$, $\Gamma_m$): Where eigenvalues cross $-1$ on the unit circle (period-doubling bifurcations).
2. One Krein collision (envelope) curve ($\Gamma_k$): Corresponds to the onset of hyperbolicity—eigenvalues collide and depart the unit circle.

The main strong results include:

• Monotonicity of the $\omega$-index: For fixed $e$, the $-1$-index $i_{-1}(\gamma_{\beta,e})$ is non-increasing as a function of $\beta$, taking values $2,1,0$ across the regions separated by the two $-1$-degeneracy curves.
• Non-degeneracy: For all $(\beta,e)\in(0,9]\times[0,1)$, the monodromy matrix is non-degenerate (i.e., no eigenvalue equals $1$ except at $\beta = 0$), precluding certain types of bifurcations.
• Classification of Stability Domains:
  • In the regime $0<\beta<\Gamma_s(e)$, solutions are elliptic-elliptic (strongly linearly stable).
  • For $\Gamma_s(e)<\beta<\Gamma_m(e)$, solutions are elliptic-hyperbolic (linearly unstable).
  • Between $\Gamma_m(e)$ and $\Gamma_k(e)$, the solutions regain stability (elliptic-elliptic), transitioning to hyperbolicity for $\beta>\Gamma_k(e)$.
• Asymptotic and Symmetry Properties: The curves demonstrate analyticity in $e$, bifurcate from $\beta=3/4$ at $e=0$ (with explicitly computed tangents), and both $-1$ degeneracy and Krein curves tend toward $(\beta,e)=(0,1)$ as $e\to 1$.

Analysis of Limiting and Boundary Cases

• Boundary $e \to 1$ (high eccentricity): The orbits become hyperbolic (unstable) except possibly near $\beta=6$, in agreement with numerical findings.
• Degeneracy for equal masses ($\beta=9$): The monodromy matrix is always hyperbolic, with all eigenvalues away from the unit circle except for positive, off-unit real eigenvalues.

Technical Contributions

• The work provides explicit characterization of monodromy matrices' normal forms in all stability regimes, including strong justification for all bifurcation phenomena via index-theoretic arguments.
• Results demonstrate that the index theory not only identifies the boundaries but quantifies the precise change in multiplicity and nature of degeneracies at the critical curves. The existence and analytic structure of these curves are established for all $\omega$ on the unit circle, revealing a foliation structure in the non-hyperbolic region.
• The established symmetry (in $e$) of the degeneracy curves is explicitly proved.

Numerical and Previous Literature Connections

• The analytic curves match with bifurcation diagrams obtained in prior numerical investigations (Martínez–Samà–Simó; see [MSS1], [MSS2]), as well as with partial results obtained via perturbative or blow-up methods.
• The paper explicitly recovers classical stability bounds (Gascheau, Routh) in the $e=0$ case and validates all known small-$e$ expansions.

Implications and Future Directions

The established global analytical framework enables the precise demarcation of stable and unstable regions for Lagrangian elliptic configurations for arbitrary masses and eccentricities. This advances the understanding of the dynamical landscape in celestial mechanics, with direct implications for both long-term stability of configurations and quantification of bifurcations leading to instability.

The analytic methods developed (Maslov-type index, operator-theoretic monotonicity and non-degeneracy arguments, variational Morse index connection) are robust and generalizable. The setup can be extended to more general $n$-body problems, including potentially higher-dimensional or non-Newtonian interactions, and systems with periodic coefficients beyond the classical three-body problem.

Further directions include:

• Rigorous asymptotic analysis near special mass parameters (e.g., $\beta=6$ at high eccentricity, where unique tangencies and degeneracies arise).
• Comprehensive investigation of the structure and connectivity of the "complex-saddle" (CS) domain.
• Extension of the analysis to spatial Lagrangian solutions and to other central configuration families.
• Exploitation of the $\omega$-index foliation for the detection of subtle bifurcation phenomena.

Conclusion

This work provides a comprehensive, mathematically rigorous platform for analyzing the linear stability of elliptic Lagrangian solutions in the planar three-body problem, linking spectral properties directly to system parameters $(\beta,e)$. The analytic approach enables the global description of stability transitions, the enumeration of bifurcation scenarios, and the validation of numerically observed phenomena. The developed framework marks a substantial advance in index-theoretic applications to periodic Hamiltonian systems and has clear potential for impact within celestial mechanics and dynamical systems theory (1206.6162).