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Normal forms for strong magnetic systems on surfaces: trapping regions and rigidity of Zoll systems

Part of: Variational problems in infinite-dimensional spaces Hamiltonian and Lagrangian mechanics Smooth dynamical systems: general theory

Published online by Cambridge University Press:  22 March 2021

LUCA ASSELLE Open the ORCID record for LUCA ASSELLE [Opens in a new window]  and
GABRIELE BENEDETTI Open the ORCID record for GABRIELE BENEDETTI [Opens in a new window]
LUCA ASSELLE
Affiliation:
Justus Liebig Universität Giessen, Mathematisches Institut, Arndtstrasse 2, 35392 Giessen, Germany (e-mail: luca.asselle@math.uni-giessen.de)
GABRIELE BENEDETTI*
Affiliation:
Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
*
e-mail: gbenedetti@mathi.uni-heidelberg.de
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Abstract

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We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings.

Keywords

invariant toriKAM theorymagnetic flowsZoll systems

MSC classification

Primary: 70H12: Periodic and almost periodic solutions
Secondary: 37C55: Periodic and quasiperiodic flows and diffeomorphisms 70H08: Nearly integrable Hamiltonian systems, KAM theory 70H11: Adiabatic invariants 70H14: Stability problems 58E10: Applications to the theory of geodesics (problems in one independent variable)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 1871 - 1897
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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