On GIT quotients of the symplectic group, stability and bifurcations of symmetric orbits

TL;DR

This paper develops a topological framework to analyze the stability and bifurcations of symmetric orbits in mechanical systems with antisymplectic symmetry, extending Krein theory to hyperbolic cases and providing tools for space mission design.

Contribution

It introduces a topological obstruction method using GIT quotients of the symplectic group to study symmetric orbit bifurcations, extending Krein theory beyond elliptic orbits.

Findings

Enumerates and describes connected components of the complement of bifurcation loci.

Provides a topological obstruction framework for orbit stability analysis.

Extends Krein theory to hyperbolic symmetric orbits.

Abstract

We provide topological obstructions to the existence of orbit cylinders of symmetric orbits, for mechanical systems preserved by antisymplectic involutions (e.g. the restricted three-body problem). Such cylinders induce continuous paths which do not cross the bifurcation locus of suitable GIT quotients of the symplectic group, which are branched manifolds whose topology provide the desired obstructions. Namely, the complement of the corresponding loci consist of several connected components which we enumerate and explicitly describe; by construction these cannot be joined by a path induced by an orbit cylinder. Our construction extends the notions from Krein theory (which only applies for elliptic orbits), to allow also for the case of symmetric orbits which are hyperbolic. This gives a general theoretical framework for the study of stability and bifurcations of symmetric orbits, with a view towards practical and numerical implementations within the context of space mission design. We shall explore this in upcoming work.