Reduction by invariants, stratifications, foliations, fibrations and relative equilibria, a short survey

TL;DR

This survey explores reduction techniques for Hamiltonian systems with symmetries, focusing on invariants, stratifications, foliations, fibrations, and stability of relative equilibria, highlighting the structure of the reduced phase space and orbit space.

Contribution

It synthesizes various results on reduction by invariants, describing the stratification, foliation, and fibration of phase and orbit spaces, and their implications for relative equilibria.

Findings

Orbit space stratification matches Thom-Boardman stratification.

Each stratum in the orbit space is foliated with symplectic leaves.

Orbit space is fibred into reduced phase spaces with symplectic leaf stratification.

Abstract

In this note we will consider reduction techniques for Hamiltonian systems that are invariant under the action of a compact Lie group $G$ acting by symplectic diffeomorphisms, and the related work on stability of relative equilibria. We will focus on reduction by invariants in which case it is possible to describe a reduced phase space within the orbit space by constructing an orbit map using a Hilbert basis of invariants for the symmetry group $G$. Results considering the stratification, foliation and fibration of the phase space and the orbit space are considered. Finally some remarks are made concerning relative equilibria and bifurcations of periodic solutions. We will combine results from a wide variety of papers. We obtain that for the orbit space the orbit type stratification coincides with the Thom-Boardman stratification and each stratum is foliated with symplectic leaves. Furthermore the orbit space is fibred into reduced phase spaces and each reduced phase space has a orbit type stratification, where each orbit type stratum is a symplectic leaf.