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Examples of integrable and non-integrable systems on singular symplectic manifolds
Delshams Valdés, Amadeu; Miranda Galcerán, Eva; Kiesenhofer, Anna
Universitat Politècnica de Catalunya. Departament de Matemàtiques; Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics; Universitat Politècnica de Catalunya. GEOMVAP - Geometria de Varietats i Aplicacions
We present a collection of examples borrowed from celes- tial mechanics and projective dynamics. In these examples symplectic structures with singularities arise naturally from regularization trans- formations, Appell's transformation or classical changes like McGehee coordinates, which end up blowing up the symplectic structure or lower- ing its rank at certain points. The resulting geometrical structures that model these examples are no longer symplectic but symplectic with sin- gularities which are mainly of two types: b m -symplectic and m -folded symplectic structures. These examples comprise the three body prob- lem as non-integrable exponent and some integrable reincarnations such as the two xed-center problem. Given that the geometrical and dy- namical 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.
Àrees temàtiques de la UPC::Matemàtiques i estadística
Symplectic manifolds
Topologia algebraica
Article - Draft

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