Nous étudions les cercles non résonants pour des champs magnétiques forts sur une surface fermée connexe orientée, et montrer comment celles-ci peuvent être utilisées pour prouver l’existence de régions piégeantes et de géodésiques magnétiques périodiques avec une vitesse faible prescrite. En corollaire, il existe une infinité de géodésiques magnétiques périodiques de petite vitesse arbitraire dans les cas suivants : i) la surface n’est pas la sphère, ii) le champ magnétique s’annule quelque part.
We study non-resonant circles for strong magnetic fields on a closed, connected, oriented surface and show how these can be used to prove the existence of trapping regions and of periodic magnetic geodesics with prescribed low speed. As a corollary, there exist infinitely many periodic magnetic geodesics for every low speed in the following cases: i) the surface is not the two-sphere, ii) the magnetic field vanishes somewhere.
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Mots clés : Magnetic systems, KAM tori, periodic orbits, trapping regions
@article{AHL_2022__5__1191_0, author = {Asselle, Luca and Benedetti, Gabriele}, title = {Non-resonant circles for strong magnetic fields on surfaces}, journal = {Annales Henri Lebesgue}, pages = {1191--1211}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.147}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ahl.147/} }
TY - JOUR AU - Asselle, Luca AU - Benedetti, Gabriele TI - Non-resonant circles for strong magnetic fields on surfaces JO - Annales Henri Lebesgue PY - 2022 SP - 1191 EP - 1211 VL - 5 PB - ÉNS Rennes UR - http://www.numdam.org/articles/10.5802/ahl.147/ DO - 10.5802/ahl.147 LA - en ID - AHL_2022__5__1191_0 ER -
Asselle, Luca; Benedetti, Gabriele. Non-resonant circles for strong magnetic fields on surfaces. Annales Henri Lebesgue, Tome 5 (2022), pp. 1191-1211. doi : 10.5802/ahl.147. http://www.numdam.org/articles/10.5802/ahl.147/
[AAB + 17] The multiplicity problem for periodic orbits of magnetic flows on the 2-sphere, Adv. Nonlinear Stud., Volume 1 (2017), pp. 1-17 | MR | Zbl
[AB15] Infinitely many periodic orbits in non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 2, pp. 1525-1545 | DOI | MR | Zbl
[AB17] On the periodic motions of a charged particle in an oscillating magnetic flows on the two-torus, Math. Z., Volume 286 (2017) no. 3-4, pp. 843-859 | DOI | Zbl
[AB21] Normal forms for strong magnetic fields on surfaces: trapping regions and rigidity of Zoll systems, Ergodic Theory Dyn. Syst. (2021), pp. 1-27 (online first) | DOI | Zbl
[AKN06] Mathematical aspects of classical and celestial mechanics, Encyclopaedia of Mathematical Sciences, 3, Springer, 2006 (also part of Dynamical systems Vol. 3], translated from the Russian original by E. Khukhro) | DOI | Zbl
[ALPS10] Motion of charged particles in magnetic fields created by symmetric configurations of wires, Physica D, Volume 239 (2010) no. 10, pp. 654-674 | DOI | MR | Zbl
[AM17] Dynamical convexity and elliptic periodic orbits for Reeb flows, Math. Ann., Volume 369 (2017) no. 1-2, pp. 331-386 | DOI | MR | Zbl
[AM19] On Tonelli periodic orbits with low energy on surfaces, Trans. Am. Math. Soc., Volume 371 (2019) no. 5, pp. 3001-3048 | DOI | MR | Zbl
[AMMP17] Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level, J. Eur. Math. Soc., Volume 19 (2017) no. 2, pp. 551-579 | DOI | MR | Zbl
[AMP15] On the existence of three closed magnetic geodesics for subcritical energies, Comment. Math. Helv., Volume 90 (2015) no. 1, pp. 155-193 | DOI | MR | Zbl
[Arn61] Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, Volume 138 (1961), pp. 255-257 | MR | Zbl
[Arn78] Mathematical methods of classical mechanics, Graduate Texts in Mathematics, 60, Springer, 1978 | Zbl
[Arn97] Remarks concerning 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., Volume 216 (1997) no. 1, pp. 3-13 (Dynamical systems and related topics: collected papers in honor of sixtieth birthday of academician Dmitrii Viktorovich Anosov) | Zbl
[AS67] Certain smooth ergodic systems, Usp. Mat. Nauk, Volume 22 (1967) no. 5 (137), pp. 107-172 | MR
[Ben16a] The contact property for symplectic magnetic fields on , Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 3, pp. 682-713 | DOI | MR | Zbl
[Ben16b] Magnetic Katok examples on the two-sphere, Bull. Lond. Math. Soc., Volume 48 (2016) no. 5, pp. 855-865 | DOI | MR | Zbl
[Ben16c] On closed orbits for twisted autonomous Tonelli Lagrangian flows, Publ. Mat. Urug., Volume 16 (2016), pp. 41-79 | MR | Zbl
[Bra70] Particle motions in a magnetic field, J. Differ. Equations, Volume 8 (1970), pp. 294-332 | DOI | MR
[BS94] Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field, Nonlinearity, Volume 7 (1994) no. 1, pp. 281-303 | DOI | MR | Zbl
[Cas01] The motion of a charged particle on a Riemannian surface under a non-zero magnetic field, J. Differ. Equations, Volume 171 (2001) no. 1, pp. 110-131 | DOI | MR | Zbl
[CMP04] Periodic orbits for exact magnetic flows on surfaces, Int. Math. Res. Not., Volume 2004 (2004) no. 8, pp. 361-387 | DOI | MR | Zbl
[GG09] Periodic orbits of twisted geodesic flows and the Weinstein-Moser theorem, Comment. Math. Helv., Volume 84 (2009) no. 4, pp. 865-907 | DOI | MR | Zbl
[GG15] The Conley conjecture and beyond, Arnold Math. J., Volume 1 (2015) no. 3, pp. 299-337 | DOI | MR | Zbl
[GGM15] On the Conley conjecture for Reeb flows, Int. J. Math., Volume 26 (2015) no. 7, 1550047 | MR | Zbl
[Gin87] New generalizations of Poincaré’s geometric theorem, Funkts. Anal. Prilozh., Volume 21 (1987) no. 2, p. 16-22, 96 | Zbl
[Gin96] On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, Contact and symplectic geometry (Thomas, C. B., ed.) (Publications of the Newton Institute), Volume 8, Cambridge University Press, 1996, pp. 131-148 | MR | Zbl
[Gol01] Symplectic twist maps, Advanced Series in Nonlinear Dynamics, 18, World Scientific, 2001 (Symplectic twist maps. Global variational techniques) | DOI | MR | Zbl
[HM03] Plasma confinement, Dover books on physics, Courier Corporation, 2003, 455 pages
[Mir07] Positive topological entropy for magnetic flows on surfaces, Nonlinearity, Volume 20 (2007) no. 8, pp. 2007-2031 | DOI | MR | Zbl
[Mos62] On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl., Volume 1962 (1962), pp. 1-20 | MR | Zbl
[MS98] Introduction to symplectic topology, Oxford Mathematical Monographs, Clarendon Press; Oxford University Press, 1998 | Zbl
[Taĭ92] Closed extremals on two-dimensional manifolds, Usp. Mat. Nauk, Volume 47 (1992) no. 2(284), p. 143-185, 223 | MR | Zbl
[Tru96] Trajectoires bornées d’une particule soumise à un champ magnétique symétrique linéaire, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 64 (1996) no. 2, pp. 127-154 | MR | Zbl
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