Symplectic approach to global stability
Verónica Errasti Díez, Jordi Gaset Rifà, Manuel Lainz
TL;DR
The paper addresses global stability for dynamical systems in a symplectic setting, with a focus on ghost-ridden Hamiltonian systems where the Hamiltonian is unbounded. It introduces confining functions and momentum-map techniques to establish $G1$ and $G2$ stability, and develops results for systems with multiple conserved quantities as well as integrable systems, including Liouville-Mineur-Arnold-type scenarios. Key contributions include criteria showing that a confining momentum map implies $G1$ stability, a demonstration that nonconfining momentum maps preclude confining combinations of conserved quantities, and a detailed analysis of integrable systems that yields precise stability conditions in terms of level-set geometry and complete symmetries. The work also formalizes ghost-ridden systems and discusses the role of bi-Hamiltonian structures, outlining future directions such as proving global stability without conserved quantities and extending the theory to noncomplete symmetries. Overall, the framework provides a geometric, testable route to proving global stability in complex Hamiltonian models with potential physical applications in ghost-ridden dynamics.
Abstract
We present a new approach to the problem of proving global stability, based on symplectic geometry and with a focus on systems with several conserved quantities. We also provide a proof of instability for integrable systems whose momentum map is everywhere regular. Our results take root in the recently proposed notion of a confining function and are motivated by ghost-ridden systems, for whom we put forward the first geometric definition.