Existence of a smooth Hamiltonian circle action near parabolic orbits

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 ar...

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Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Kudryavtseva, Elena, Martynchuk, Nikolay
Format: Article
Language:English
Subjects:
Orbital stability
Orbits
Perturbation
Structural stability
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2106.04838