Linear stability of the elliptic relative equilibria for the restricted $4$-body problem: the Euler case

TL;DR

This paper investigates the linear stability of elliptic relative equilibria in a restricted 4-body problem with collinear primaries, using symplectic reduction and Maslov index, providing stability conditions based on mass parameters.

Contribution

It introduces a symplectic reduction approach to the restricted 4-body problem and analyzes stability via the Maslov index, connecting it to elliptic Lagrangian solutions.

Findings

Derived stability conditions for symmetric cases.

Established relationship between restricted 4-body and elliptic Lagrangian solutions.

Provided numerical stability criteria based on mass parameters.

Abstract

In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{R}^2$. We obtain the symplectic reduction to the general restricted $N$-body problem. By analyzing the relationship between this restricted $4$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $4$-body problem by the $\omega$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters for the symmetric cases.