Persistence of stationary motion under explicit symmetry breaking perturbation

TL;DR

This paper develops a geometric framework to analyze how stationary motions in symmetric Hamiltonian systems persist or change when explicit symmetry-breaking perturbations are introduced, providing bounds based on topological invariants.

Contribution

It introduces a geometric approach to quantify the persistence of equilibria under symmetry-breaking perturbations in equivariant Hamiltonian systems, using equivariant Lyusternik-Schnirelmann category.

Findings

Provides a lower bound for the number of persistent equilibria and relative equilibria.

Connects the persistence of stationary motions to topological invariants of the symmetry group.

Offers a method to predict system behavior under symmetry-breaking perturbations.

Abstract

Explicit symmetry breaking occurs when a dynamical system having a certain symmetry group is perturbed in a way that the perturbation preserves only some symmetries of the original system. We give a geometric approach to study this phenomenon in the setting of equivariant Hamiltonian systems. A lower bound for the number of orbits of equilibria and orbits of relative equilibria which persist after a small perturbation is given. This bound is given in terms of the equivariant Lyusternik-Schnirelmann category of the group orbit.