Cotangent models for integrable systems

Abstract

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold. For this latter class of Poisson structures we define a twisted cotangent model. The equivalence with this twisted cotangent model is given by an action-angle theorem recently proved in [KMS16]. This viewpoint of cotangent models provides a new machinery to construct examples of integrable systems, which are especially valuable in the $b$-symplectic case where not many sources of examples are known. At the end of the paper we introduce non-degenerate singularities as lifted cotangent models on $b$-symplectic manifolds and discuss some generalizations of these models to general Poisson manifolds.

25 pages; final version to appear at Communications in Mathematical Physics

References in corpus (2)

• Examples of integrable and non-integrable systems on singular symplectic manifolds
• Action-angle variables and a KAM theorem for b-Poisson manifolds

Cited by in corpus (15)

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• Reduction theory for singular symplectic manifolds and singular forms on moduli spaces
• $b$-Structures on Lie groups and Poisson reduction
• Constructions of b-semitoric systems
• A $b$-symplectic slice theorem
• Singular cotangent models and complexity in fluids with dissipation
• Rigidity of cotangent lifts and integrable systems
• A counterexample to the singular Weinstein conjecture
• Integrable systems on singular symplectic manifolds: From local to global
• The Arnold conjecture for singular symplectic manifolds
• E-structures and almost regular Poisson manifolds
• Geometric quantization via cotangent models