On the Stability of Lagrange Relative Equilibrium in The Planar Three-body Problem

TL;DR

This paper proves that Lagrange relative equilibrium in the planar three-body problem is stable in measure and exponentially stable for most mass configurations, using a new coordinate system to handle degeneracies.

Contribution

It introduces a new coordinate system to analyze stability, proving measure and exponential stability of Lagrange equilibrium under spectral stability, with near-complete measure of stability.

Findings

Lagrange relative equilibrium is stable in measure under spectral stability.

Abundant KAM tori or quasi-periodic solutions exist near equilibrium.

Exponential stability holds for almost all mass choices.

Abstract

Since the strong degeneracies present in the N-body problem, even in the basic case of the planar three-body problem, nobody inspects the problem of nonlinear stability of Lagrange relative equilibrium. We introduce a new coordinate system to reduce degeneracies according to intrinsic symmetrical characteristic of the N-body problem, then we prove that Lagrange relative equilibrium is stable in the sense of measure, provided it is spectrally stable and except six special resonant cases. Indeed, under this condition, there are abundant KAM invariant tori or quasi-periodic solutions near Lagrange relative equilibrium. Furthermore, these tori or quasi-periodic solutions form a set whose relative measure rapidly tends to 1. We also prove that Lagrange relative equilibrium is exponentially stable for almost every choice of masses in the sense of measure, provided it is spectrally stable; and topologically, this is also right for a large open subset of spectrally stable space of masses.