Examples of integrable and non-integrable systems on singular symplectic manifolds
| Autor: | Eva Miranda, Anna Kiesenhofer, Amadeu Delshams |
|---|---|
| Přispěvatelé: | Departament de Matemàtiques [Barcelona] (UAB), Universitat Autònoma de Barcelona (UAB), Observatoire de Paris, Université Paris sciences et lettres (PSL), Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions, Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC |
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
General Physics and Astronomy Dynamical Systems (math.DS) 01 natural sciences Matemàtiques i estadística::Equacions diferencials i integrals [Àrees temàtiques de la UPC] Symplectic vector space 0103 physical sciences FOS: Mathematics 0101 mathematics Mathematics - Dynamical Systems [MATH]Mathematics [math] Symplectomorphism Moment map Integral equations Mathematics::Symplectic Geometry Mathematical Physics Mathematics Symplectic manifold Symplectic group 010102 general mathematics 3-body problem Symplectic representation Equacions integrals Symplectic matrix Algebra b-Poisson manifolds Mathematics - Symplectic Geometry Integrable systems Symplectic Geometry (math.SG) 010307 mathematical physics Geometry and Topology Symplectic geometry |
| Zdroj: | Journal of Geometry and Physics Journal of Geometry and Physics, Elsevier, 2015, 115, pp.89-97. ⟨10.1016/j.geomphys.2016.06.011⟩ Recercat. Dipósit de la Recerca de Catalunya instname UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| ISSN: | 0393-0440 |
| DOI: | 10.1016/j.geomphys.2016.06.011⟩ |
| Popis: | We present a collection of examples borrowed from celestial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization transformations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lowering its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with singularities which are mainly of two types: $b^m$-symplectic and $m$-folded symplectic structures. These examples comprise the three body problem as non-integrable exponent and some integrable reincarnations such as the two fixed-center problem. Given that the geometrical and dynamical properties of $b^m$-symplectic manifolds and folded symplectic manifolds are well-understood [GMP, GMP2, GMPS, KMS, Ma, CGP, GL,GLPR, MO, S, GMW], we envisage that this new point of view in this collection of examples can shed some light on classical long-standing problems concerning the study of dynamical properties of these systems seen from the Poisson viewpoint. 14 pages |
| Databáze: | OpenAIRE |
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