Existence of a smooth Hamiltonian circle action near parabolic orbits

TL;DR

This paper proves that parabolic orbits in two-degree-of-freedom integrable systems have smooth Hamiltonian circle actions that persist under small perturbations, establishing their structural stability and smooth equivalence to a standard model.

Contribution

It demonstrates the existence and persistence of smooth Hamiltonian circle actions near parabolic orbits, a new result in the study of integrable systems.

Findings

Parabolic orbits admit smooth Hamiltonian circle actions.

These actions are stable under small integrable perturbations.

All parabolic orbits are smoothly equivalent to a standard model.

Abstract

We show that every parabolic orbit of a two-degree of freedom integrable system admits a $C^\infty$-smooth Hamiltonian circle action, which is persistent under small integrable $C^\infty$ perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the integrals of motion is Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.