Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory

TL;DR

This paper analyzes the linear stability of Lagrangian circular orbits in a generalized three-body problem with various potentials, identifying stability regions and Morse index jumps using advanced index theory.

Contribution

It extends stability analysis to broader singular potentials and computes Morse indices, revealing stability boundaries via index theory.

Findings

Identifies stability regions in terms of potential and mass parameters.

Computes Morse indices of orbits and their iterates.

Finds stability boundary as the envelope of Morse index jump curves.

Abstract

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter $\beta$ and on the eccentricity $e$ of the orbit. We consider only the circular case ($e = 0$) but under the action of a broader family of singular potentials: $\alpha$-homogeneous potentials, for $\alpha \in (0,2)$, and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter $\alpha$ and the mass parameter $\beta$, then we compute the Morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the Morse indices of the iterates jump. In order to conduct our analysis we rely on a Maslov-type index theory devised and developed by Y.~Long, X.~Hu and S.~Sun; a key role is played by an appropriate index theorem and by some precise computations of suitable Maslov-type indices.