Examples of integrable and non-integrable systems on singular symplectic manifolds

Abstract

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

References in corpus (3)

• Cotangent models for integrable systems
• Action-angle variables and a KAM theorem for b-Poisson manifolds
• Global instability in the elliptic restricted three body problem

Cited by in corpus (16)

• Cotangent models for integrable systems
• Desingularizing $b^m$-symplectic structures
• Non-commutative integrable systems on $b$-symplectic manifolds
• Global instability in the elliptic restricted three body problem
• On the volume elements of a manifold with transverse zeroes
• $b$-Structures on Lie groups and Poisson reduction
• Constructions of b-semitoric systems
• Singular cotangent models and complexity in fluids with dissipation
• b-Contact Structures on Tentacular Hyperboloids
• On the singular Weinstein conjecture and the existence of escape orbits for $b$-Beltrami fields
• Hamiltonian facets of classical gauge theories on $E$-manifolds
• Regularization of central forces with damping in two and three-dimensions
• The Arnold conjecture for singular symplectic manifolds
• The singular Weinstein conjecture
• Integrable systems on singular symplectic manifolds: From local to global
• Zeroth Poisson homology, foliated cohomology and perfect Poisson manifolds