Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{\infty}$$\end{document}-smooth Hamiltoniancircle action, which is persistent under small integrable \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{\infty}$$\end{document} perturbations.We deduce from this result the structural stability of parabolic orbits and show that they are all smoothlyequivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing thatevery symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on theconnected components of the common level sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Regular and Chaotic Dynamics Springer Journals

Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

, Volume 26 (6) – Nov 1, 2021
10 pages

/lp/springer-journals/existence-of-a-smooth-hamiltonian-circle-action-near-parabolic-orbits-tXwFIZWEcC
Publisher
Springer Journals
ISSN
1560-3547
eISSN
1468-4845
DOI
10.1134/s1560354721060101
Publisher site
See Article on Publisher Site

Abstract

We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{\infty}$$\end{document}-smooth Hamiltoniancircle action, which is persistent under small integrable \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C^{\infty}$$\end{document} perturbations.We deduce from this result the structural stability of parabolic orbits and show that they are all smoothlyequivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing thatevery symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on theconnected components of the common level sets.

Journal

Regular and Chaotic DynamicsSpringer Journals

Published: Nov 1, 2021

Keywords: Liouville integrability; parabolic orbit; circle action; structural stability; normal forms.

References

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