[Paper Review] Jacobi stability analysis of the classical restricted three body problem

Cristina Blaga, Paul A. Blaga+1 more|arXiv (Cornell University)|Apr 6, 2021

Advanced Differential Geometry Research7 references4 citations

TL;DR

This paper applies Kosambi-Cartan-Chern (KCC) theory to analyze Jacobi stability in the classical circular restricted three-body problem. Using differential geometric methods to study trajectory deviations, it finds that all five Lagrangian equilibrium points—L₁, L₂, L₃, L₄, and L₅—are Jacobi unstable for all values of the mass parameter μ₂, contrasting with linear (Lyapunov) stability which shows L₄ and L₅ are stable under specific mass conditions.

ABSTRACT

The circular restricted three body problem, which considers the dynamics of an infinitesimal particle in the presence of the gravitational interaction with two massive bodies moving on circular orbits about their common center of mass, is a very useful model for investigating the behavior of real astronomical objects in the Solar System. In such a system, there are five Lagrangian equilibrium points, and one important characteristic of the motion is the existence of linearly stable equilibria at the two equilibrium points that form equilateral triangles with the primaries, in the plane of the primaries' orbit. We analyze the stability of motion in the restricted three body problem by using the concept of Jacobi stability, as introduced and developed in the Kosambi-Cartan-Chern (KCC) theory. The KCC theory is a differential geometric approach to the variational equations describing the deviation of the whole trajectory of a dynamical system with respect to the nearby ones. We obtain the general result that, from the point of view of the KCC theory and of Jacobi stability, all five Lagrangian equilibrium points of the restricted three body problem are unstable.

Motivation & Objective

• To investigate the stability of the five Lagrangian equilibrium points in the circular restricted three-body problem using Jacobi stability from KCC theory.
• To compare Jacobi stability with classical linear (Lyapounov) stability, especially regarding the stability of L₄ and L₅ points.
• To develop and apply geometric methods based on KCC theory to analyze trajectory deviations in nonlinear dynamical systems.
• To determine whether the stability of equilibrium points is independent of mass parameters under the Jacobi stability framework.
• To provide a geometric characterization of robustness against perturbations across entire trajectories, not just local equilibria.

Proposed method

• Formalize the restricted three-body problem as a system of second-order nonlinear ordinary differential equations in a rotating frame.
• Apply the Kosambi-Cartan-Chern (KCC) theory to geometrically describe the system by introducing a non-linear connection and a Berwald-type connection.
• Construct the five KCC invariants, with the second invariant (deviation curvature tensor) used to assess Jacobi stability.
• Compute the eigenvalues of the deviation curvature tensor at each Lagrangian point to determine stability properties.
• Analyze the characteristic equation and its discriminant and product of roots for L₁, L₂, L₃ (collinear) and L₄, L₅ (triangular) points.
• Use the sign and nature of the eigenvalues (real and negative product) to conclude Jacobi instability for all five points.

Experimental results

Research questions

• RQ1 Are all five Lagrangian equilibrium points in the restricted three-body problem Jacobi stable according to KCC theory?
• RQ2 How does Jacobi stability differ from classical linear (Lyapounov) stability in the context of the restricted three-body problem?
• RQ3 Is the Jacobi instability of the Lagrangian points dependent on the mass parameter μ₂?
• RQ4 Can the KCC theory provide a geometric characterization of trajectory robustness independent of local linearization?
• RQ5 What is the role of the deviation curvature tensor in determining the stability of entire trajectories in nonlinear dynamical systems?

Key findings

• All five Lagrangian equilibrium points (L₁, L₂, L₃, L₄, L₅) in the restricted three-body problem are found to be Jacobi unstable for all values of the mass parameter μ₂.
• For the collinear points L₁, L₂, L₃, the characteristic equation has real eigenvalues with a negative product, confirming instability via the KCC second invariant.
• The discriminant of the characteristic equation for collinear points is a nonzero perfect square, indicating real and distinct roots, consistent with instability.
• The product of the eigenvalues for collinear points is P = -2(μ₁/r₁³ + μ₂/r₂³)², a negative, nonzero real number, confirming instability.
• The triangular points L₄ and L₅, which are linearly stable for μ₂ ≤ (27 - √621)/54 ≈ 0.0385, are still Jacobi unstable for all μ₂ values.
• Jacobi stability is independent of the mass parameter μ₂, unlike linear stability, which depends on specific numerical thresholds.