Integrable systems on b-symplectic manifolds
The study of b-symplectic manifolds was initiated in 2012 by the works of Victor Guillemin, Eva Miranda and Ana Rita Pires (Adv. Math. 264 (2014), 864¿896). These manifolds, which can be understood as symplectic manifolds with singularities, have since then become the object of intense study. In the...
| Autor: | |
|---|---|
| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2016 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/457145 |
| Acceso en línea: | http://hdl.handle.net/10803/457145 https://dx.doi.org/10.5821/dissertation-2117-111236 |
| Access Level: | acceso abierto |
| Palabra clave: | Àrees temàtiques de la UPC::Matemàtiques i estadística 514 |
| Sumario: | The study of b-symplectic manifolds was initiated in 2012 by the works of Victor Guillemin, Eva Miranda and Ana Rita Pires (Adv. Math. 264 (2014), 864¿896). These manifolds, which can be understood as symplectic manifolds with singularities, have since then become the object of intense study. In the language of Poisson tensors, a b-symplectic manifold is a manifold $M^{2n}$ with a Poisson tensor $\Pi$ such that $\Pi^n$ vanishes transversally to the zero section of the bundle $¿^{2n}TM$. This thesis contributes important results about the dynamics of b-symplectic manifolds. After reviewing the general theory of Poisson and $b$-symplectic manifolds we present the definitions of integrable systems on these manifolds. The main results of this thesis are the action-angle coordinate theorems for commutative and non-commutative b-integrable systems [KMS, KM2], which state the existence of invariant "Liouville" tori on the singular set of the b-symplectic manifold, in perfect analogy to the symplectic case. We go on to present a cotangent model of this result, identifying a neighborhood of such a Liouville torus with a certain cotangent lift of a torus action [KM1]. These models also allow us to construct examples of b-integrable systems using torus actions as a starting point. The existence of action-angle coordinates motivates us to explore an analogue of the classical stability result for symplectic manifolds known as KAM theory. We prove a result that shows stability of a large number of invariant tori under certain perturbations. Finally, we present several examples of singular symplectic structures that arise naturally as the result of non-canonical transformations used in regularization of singularities in celestial mechanics [DKM]. |
|---|