[Paper Review] Action-angle coordinates and KAM theory for singular symplectic manifolds

Eva Miranda, Arnau Planas|arXiv (Cornell University)|Dec 31, 2022

Advanced Differential Equations and Dynamical Systems4 citations

TL;DR

This paper establishes KAM theory for $b^m$-symplectic manifolds—singular symplectic structures with transverse hypersurfaces—by developing action-angle coordinates and proving a KAM theorem that guarantees the persistence of quasi-periodic tori under small perturbations. The key contribution is the first perturbation theory for integrable systems on Poisson manifolds beyond the symplectic category, with applications to celestial mechanics and fluid dynamics via desingularization and $b$-contact structures.

ABSTRACT

This monograph explores classification and perturbation problems for integrable systems on a class of Poisson manifolds called $b^m$-Poisson manifolds. Even if the class of $b^m$-Poisson manifolds is not ample enough to represent general Poisson manifolds, this investigation can be seen as a first step for the study of perturbation theory for general Poisson manifolds. We prove an action-angle coordinate and a KAM theorem for $b^m$-Poisson manifolds which improves the one obtained for $b$-Poisson manifolds for $m=1$ in [KMS16]. As an outcome of this result together with the extension of the desingularization techniques of Guillemin-Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem for folded symplectic manifolds. We also obtain a new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface. In several problems in celestial mechanics, this distinguished hypersurface can be the line at infinity or can represent the collision set.

Motivation & Objective

• To extend KAM theory to $b^m$-symplectic manifolds, a class of singular symplectic manifolds with transverse critical hypersurfaces, beyond the classical symplectic setting.
• To develop an action-angle coordinate formalism for $b^m$-integrable systems, generalizing the classical Arnold-Liouville-Mineur theorem to singular Poisson structures.
• To establish a KAM theorem for $b^m$-symplectic manifolds that ensures the persistence of invariant tori under small perturbations, enabling the study of quasi-periodic dynamics.
• To apply the theory to celestial mechanics, particularly the restricted three-body problem and escape orbits, using desingularization techniques like McGehee coordinates.
• To connect $b^m$-dynamics to fluid dynamics via the $b$-Reeb-Beltrami correspondence, identifying singular periodic and escape orbits in stationary fluid flows.

Proposed method

• Introduce $b^m$-symplectic structures as generalizations of symplectic forms that are non-degenerate away from a smooth hypersurface $Z$ and admit a smooth extension via $b^m$-forms.
• Construct action-angle coordinates for $b^m$-integrable systems by defining action variables via integration of $b^m$-one-forms over cycles in the regular locus.
• Prove a $b^m$-symplectic version of the Arnold-Liouville theorem, showing that locally, $b^m$-integrable systems admit a torus action with angle coordinates on the regular level sets.
• Develop a new KAM theorem for $b^m$-symplectic manifolds by reducing the perturbation problem to solving a $b^m$-symplectomorphism equation using Nash-Moser iteration in the $b^m$-framework.
• Use desingularization techniques (e.g., McGehee transformation) to relate $b^m$-structures to classical celestial mechanics problems, such as the Kepler problem and two-fixed-center problem.
• Leverage the $b$-Reeb-Beltrami correspondence to transfer results on periodic and escape orbits from contact geometry to fluid dynamics and celestial mechanics.

Experimental results

Research questions

• RQ1 Can KAM theory be extended to $b^m$-symplectic manifolds, a class of singular Poisson manifolds where the symplectic form degenerates along a hypersurface?
• RQ2 Do $b^m$-integrable systems admit a well-defined action-angle coordinate system analogous to the classical symplectic case?
• RQ3 Under what conditions do quasi-periodic tori persist under small perturbations in $b^m$-symplectic systems?
• RQ4 How can the KAM theorem for $b^m$-symplectic manifolds be applied to locate new periodic orbits in celestial mechanics, particularly near infinity or collision singularities?
• RQ5 What is the role of $b^m$-contact structures and the $b$-Reeb-Beltrami correspondence in identifying escape orbits and singular periodic orbits in fluid dynamics and celestial systems?

Key findings

• A new KAM theorem is established for $b^m$-symplectic manifolds, proving the persistence of quasi-periodic tori under small $C^ u$-smooth perturbations in the $b^m$-category.
• Action-angle coordinates are constructed for $b^m$-integrable systems using $b^m$-forms and cycles in the regular locus, generalizing the classical Arnold-Liouville theorem.
• The McGehee transformation is shown to convert the Kepler problem into a $b^3$-contact structure, where the Liouville vector field becomes a $b^3$-vector field transverse to energy level sets.
• After desingularization, the restricted three-body problem near infinity exhibits $b^3$-contact structures with infinitely many non-trivial periodic orbits on the critical set.
• The $b$-Reeb-Beltrami correspondence implies that generic $b$-Beltrami vector fields on 3-manifolds with $N$ connected components of the critical set have at least $2N$ or infinitely many escape orbits.
• The KAM method can be used to perturb these singular periodic orbits into genuine periodic orbits in the perturbed system, extending Poincaré’s continuation method to the $b^m$-setting.