Orbital stability: analysis meets geometry

TL;DR

This paper introduces a geometric and analytical framework for analyzing the orbital stability of relative equilibria in Hamiltonian systems, emphasizing the energy-momentum method and illustrating with PDE examples.

Contribution

It develops a unified approach combining symplectic geometry and functional analysis for stability analysis of Hamiltonian systems with symmetry.

Findings

Unified geometric-analytical framework for stability

Application to Hamiltonian PDEs like NLS and wave equations

Illustrations with finite and infinite-dimensional systems

Abstract

We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the wave equation, and for the Manakov system.