2002/01/24 by George W. Patrick, Mark Roberts, Patrick, George W. +3
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Black Holes and Theoretical Physics #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG) #math-ph #math.DS #math.MP #math.SG
paper · pdf · doi:10.48550/arxiv.math/0201239
31 pages
arxiv created 2002/01/24 · openalex publication_date 2002/01/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum methods. Using a topological generalisation of Lyapunov's result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the non-Hausdorff nature of the symplectic leaf space at that point. This criterion is applied to generalise the Energy-Momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a G-stable relative equilibrium satisfies the stronger condition of being A-stable, where A is a specific group-theoretically defined subset of G which contains the momentum isotropy subgroup of the relative equilibrium.