[Paper Review] Stability of fixed points and associated relative equilibria of the $3$-body problem on $\mathbb S^1$ and $\mathbb S^2$

Florin Diacu, Juan Manuel Sánchez-Cerritos+1 more|arXiv (Cornell University)|Mar 10, 2016

Spacecraft Dynamics and Control14 references3 citations

TL;DR

This paper investigates the stability of fixed-point solutions and associated relative equilibria in the curved 3-body problem on the 1-sphere ($\mathbb{S}^1$) and 2-sphere ($\mathbb{S}^2$). It proves Lyapunov stability for fixed points and their relative equilibria on $\mathbb{S}^1$, while on $\mathbb{S}^2$, linear stability of relative equilibria depends on angular velocity: they are linearly stable if and only if $\omega^2 > \lambda_1$, where $\lambda_1$ is a configuration-dependent critical value.

ABSTRACT

We prove that the fixed points of the curved 3-body problem and their associated relative equilibria are Lyapunov stable if the solutions are restricted to $\mathbb S^1$, but unstable if the bodies are considered in $\mathbb S^2$.

Motivation & Objective

• To establish the existence and stability properties of fixed-point solutions in the curved 3-body problem on $\mathbb{S}^1$ and $\mathbb{S}^2$.
• To analyze the stability of relative equilibria associated with these fixed points, particularly focusing on the role of angular velocity on $\mathbb{S}^2$.
• To provide a general criterion for the existence of fixed points for $n > 2$ masses on $\mathbb{S}^2$, and to classify all mass triples that admit such configurations.
• To extend the classical stability framework to curved configuration spaces by analyzing the linearized dynamics on invariant subspaces.
• To compare the stability behavior of fixed points on $\mathbb{S}^1$ with classical collinear central configurations in the Newtonian $n$-body problem.

Proposed method

• Formulating the curved 3-body problem as a Hamiltonian system on the cotangent bundle of the configuration space $W = (\mathbb{S}^2)^n \setminus \Delta$, using geodesic distances and a cotangent potential involving $\cot d_{ij}$.
• Defining fixed points as critical points of the force function $V = -U$, where $U$ is the potential, ensuring zero-velocity solutions on the sphere.
• Using Jacobi-type coordinates to reduce the system and analyze stability in the reduced phase space, particularly focusing on rest points of the flow.
• Applying Sylvester’s law of inertia and matrix congruence to analyze the Hessian of the force function and determine the number of negative eigenvalues, establishing Lyapunov stability on $\mathbb{S}^1$.
• Employing a reduction technique inspired by Rick Moeckel to identify proper linear subspaces $E_2$ and $\tilde{E}$ for studying linear stability on $\mathbb{S}^2$, restricting the linearized system to $\tilde{E}$.
• Computing the eigenvalues of the linearized operator $HM^{-1} - \omega^2$ to determine the stability condition: purely imaginary eigenvalues imply linear stability, which occurs precisely when $\omega^2 > \lambda_1$.

Experimental results

Research questions

• RQ1 Under what conditions do fixed-point solutions exist for the curved 3-body problem on $\mathbb{S}^1$ and $\mathbb{S}^2$?
• RQ2 Are fixed-point solutions and their associated relative equilibria Lyapunov stable on $\mathbb{S}^1$?
• RQ3 How does the angular velocity $\omega$ affect the linear stability of relative equilibria on $\mathbb{S}^2$?
• RQ4 What is the precise critical value $\lambda_1$ that determines the threshold for linear stability on $\mathbb{S}^2$?
• RQ5 How do the stability properties of fixed points on $\mathbb{S}^1$ compare to those of classical collinear central configurations in the Newtonian $n$-body problem?

Key findings

• Fixed-point solutions on $\mathbb{S}^1$ are Lyapunov stable because they correspond to local maxima of the force function $V$ in the reduced phase space.
• On $\mathbb{S}^2$, fixed-point configurations exist only if the three masses form an acute triangle on a great circle.
• The associated relative equilibria on $\mathbb{S}^2$ are linearly stable if and only if the square of the angular velocity exceeds a critical value: $\omega^2 > \lambda_1$.
• The critical value $\lambda_1$ is explicitly computed as $\lambda_1 = -\frac{m_2}{\sin^2\alpha}\frac{\sin\beta}{\sin(\alpha+\beta)\sin\alpha} - \frac{m_3}{\sin^2\beta}\frac{\sin\alpha}{\sin(\alpha+\beta)\sin\beta} - \frac{m_3}{\sin^2\beta}\frac{\sin(\alpha+\beta)}{\sin\alpha\sin\beta}$, where $\alpha, \beta$ are angles of the acute triangle.
• The linear stability on $\mathbb{S}^2$ is determined by the spectrum of the matrix $HM^{-1} - \omega^2$, with purely imaginary eigenvalues indicating stability when $\omega^2 > \lambda_1$.
• The system exhibits a fundamental difference from the classical $n$-body problem: while classical collinear configurations are stable minima of the potential on a line, fixed points on $\mathbb{S}^1$ are maxima, and stability on $\mathbb{S}^2$ is velocity-dependent rather than configuration-only.