Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases

TL;DR

This paper analyzes the linear stability of elliptic relative equilibria in restricted N-body problems with specific configurations, using symplectic reduction, Maslov index, and numerical methods to determine stability conditions.

Contribution

It provides new stability results for restricted N-body problems with collinear and polygonal configurations, including analytical and numerical stability criteria.

Findings

Stability conditions derived for N=4 based on mass parameters and symmetry.

Identification of stable and unstable regions depending on the position of the massless body.

Application of the $ ext{ω}$-Maslov index to analyze linear stability.

Abstract

In this paper, we consider the elliptic relative equilibria of the restricted $N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear central configuration or a $(1+n)$-gon central configuration. We obtain the symplectic reduction to the general restricted $N$-body problem. For the first case, by analyzing the relationship between this restricted $N$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $N$-body problem by the $\omega$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters under $N=4$ and the symmetry of the central configuration. For the second case, there exist three positions $S_1,S_2$ and $S_3$ of the massless body (up to rotations of angle $\frac{2\pi}{n}$). For ${m_0\over m}$ sufficiently large, we show that the elliptic relative equilibria is linearly unstable if the eccentricity $0\le e<e_0$ and the massless body lies at $S_1$ or $S_2$; while the elliptic relative equilibria is linear stability if the massless body lies at $S_3$.