Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T20:23:37.765Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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, 35392Giessen, Germany (e-mail: luca.asselle@math.uni-giessen.de)
GABRIELE BENEDETTI*
Affiliation:
Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 205, 69120Heidelberg, Germany
*
e-mail: gbenedetti@mathi.uni-heidelberg.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 1871 - 1897
Check for updates
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

References

Arnold, V. I.. Some remarks on flows of line elements and frames. Dokl. Akad. Nauk 138 (1961), 255–257.Google Scholar
Arnold, V. I.. Mathematical Methods of Classical Mechanics. Springer, New York, NY, 1978.CrossRefGoogle Scholar
Arnold, V. I.. Remarks on the Morse theory of a divergence-free vector field, the averaging method, and the motion of a charged particle in a magnetic field. Proc. Steklov Inst. Math. 216(1) (1997), 3–13 (Dynamical systems and related topics: collected papers in honor of sixtieth birthday of academician Dmitrii Viktorovich Anosov).Google Scholar
Asselle, L. and Benedetti, G.. Integrable magnetic flows on the two-torus: Zoll examples and systolic inequalities. J. Geom. Anal. to appear, 2020, doi:10.1007/s12220-020-00379-1.CrossRefGoogle Scholar
Asselle, L. and Lange, C.. On the rigidity of Zoll magnetic systems on surfaces. Nonlinearity 33(7) (2020), 3173–3194.CrossRefGoogle Scholar
Benedetti, G.. The contact property for magnetic flows on surfaces. PhD Thesis, University of Cambridge, 2015, doi:10.17863/CAM.16235.CrossRefGoogle Scholar
Benedetti, G. and Kang, J.. On a systolic inequality for closed magnetic geodesics on surfaces. Preprint, 2018, arXiv:1902.01262.Google Scholar
Benettin, G. and Sempio, P.. Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field. Nonlinearity 7(1) (1994), 281–303.10.1088/0951-7715/7/1/014CrossRefGoogle Scholar
Bertrand, J.. Théorème relatif au mouvement d’un point attiré vers un centre fixe. C. R. Acad. Sci. 77 (1873), 849–853.Google Scholar
Castilho, C.. The motion of a charged particle on a Riemannian surface under a non-zero magnetic field. J. Differential Equations 171(1) (2001), 110–131.CrossRefGoogle Scholar
Ginzburg, V. L.. New generalizations of Poincaré’s geometric theorem. Funktsional. Anal. i Prilozhen. 21(2) (1987), 16–22, 96.10.1007/BF01078023CrossRefGoogle Scholar
Ginzburg, V. L.. On closed trajectories of a charge in a magnetic field. An application of symplectic geometry. Contact and Symplectic Geometry (Publications of the Newton Institute, 8). Cambridge University Press, Cambridge, 1996, pp. 131–148.Google Scholar
Guillemin, V.. The Radon transform on Zoll surfaces. Adv. Math. 22(1) (1976), 85–119.CrossRefGoogle Scholar
Helffer, B., Kordyukov, Y., Raymond, N. and Vũ Ngọc, S.. Magnetic wells in dimension three. Anal. PDE 9(7) (2016), 1575–1608.CrossRefGoogle Scholar
Kudryavtseva, E. A. and Podlipaev, S. A.. Superintegrable Bertrand magnetic geodesic flows. Fundam. Prikl. Mat. 22(6) (2019), 169–182.Google Scholar
Martins, G.. The Hamiltonian dynamics of magnetic confinement in toroidal domains. Pacific J. Math. 304(2) (2020), 613–628.CrossRefGoogle Scholar
Moser, J.. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 (1962), 1–20.Google Scholar
Moser, J.. On the volume elements on a manifold. Trans. Amer. Math. Soc. 120 (1965), 286–294.CrossRefGoogle Scholar
Raymond, N. and Vũ Ngọc, S.. Geometry and spectrum in 2D magnetic wells. Ann. Inst. Fourier (Grenoble) 65(1) (2015), 137–169.CrossRefGoogle Scholar
Zoll, O.. Ueber Flächen mit Scharen geschlossener geodätischer Linien. Math. Ann. 57(1) (1903), 108–133.CrossRefGoogle Scholar