Symmetric solutions of the $n$-body problem: a numerical study of Floquet multipliers and Morse indices

TL;DR

The paper advances the numerical study of symmetric periodic orbits in the $n$-body problem by integrating group-equivariant variational methods with stability analysis. It formalizes the configuration and symmetry settings, introducing two complementary stability indicators—Floquet multipliers and the discrete Morse index—and develops discretisation schemes (point and Fourier) to compute these indicators on both fundamental domains and full orbits. Applying these methods to ten symmetric orbits (via SymOrb.jl) yields a nuanced picture of action minimisation versus linear stability, highlighting cases like the triangular Lagrange solution and the figure-eight choreography. The work provides a practical, benchmarked workflow for classifying and analysing symmetric periodic solutions, with clear implications for understanding the qualitative dynamics of the $n$-body problem.

Abstract

In this paper, we consider periodic solutions of the $n$-body problem that satisfy symmetry constraints, expressed through invariance under finite group actions. We focus on their stability properties and present algorithms specifically designed for the computation of Floquet multipliers and Morse indices. Numerical results are provided to illustrate our methods in both two and three dimensional configuration spaces, and for different choices on the number of bodies.

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Theorems & Definitions (1)

• Definition 1: Computational discrete Morse index