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Integrable systems on singular symplectic manifolds: from local to global
- Source :
- UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC)
- Publication Year :
- 2021
-
Abstract
- In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [36] and [37] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and bsymplectic forms in [36]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set Z of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z. Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Both authors are supported by the grants reference number 2017SGR932 (AGAUR) and PID2019-103849GB-I00 / AEI / 10.13039/501100011033.
- Subjects :
- Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics [Àrees temàtiques de la UPC]
Symplectic geometry
Matemàtiques i estadística [Àrees temàtiques de la UPC]
Geometria simplèctica
Sistemes dinàmics diferenciables
53 Differential geometry::53D Symplectic geometry, contact geometry [Classificació AMS]
Dynamical Systems
58 Global analysis, analysis on manifolds [Classificació AMS]
Symplectic structures
37 Dynamical systems and ergodic theory [Classificació AMS]
Integrable systems
Matemàtiques i estadística::Geometria::Geometria diferencial [Àrees temàtiques de la UPC]
Mathematics::Symplectic Geometry
Mathematical Physics
Differential Geometry
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- UPCommons. Portal del coneixement obert de la UPC, Universitat Politècnica de Catalunya (UPC)
- Accession number :
- edsair.dedup.wf.001..031a7a9814f8d64f357fe2ae51b65a69