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

TL;DR

This paper explores examples of systems in celestial mechanics that naturally lead to symplectic structures with singularities, such as $b^m$-symplectic and folded symplectic structures, providing insights into their dynamical properties.

Contribution

It introduces a collection of celestial mechanics examples modeled by singular symplectic structures, connecting classical problems with modern geometric frameworks.

Findings

Examples include the three body problem and two fixed-center problem.

Singular symplectic structures arise from regularization and coordinate transformations.

Potential to inform classical problems through the Poisson viewpoint.

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.