On the existence of periodic orbits for magnetic systems on the two-sphere

Gabriele  Benedetti; Kai  Zehmisch

doi:10.3934/jmd.2015.9.141

Article Contents

2015, Volume 9: 141-146. Doi: 10.3934/jmd.2015.9.141

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On the existence of periodic orbits for magnetic systems on the two-sphere

Gabriele Benedetti^1, and
Kai Zehmisch^1,

1.
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster

Received: March 2015

Published: June 2015

Abstract / Introduction Related Papers Cited by

Abstract

We prove that there exist periodic orbits on almost all compact regular energy levels of a Hamiltonian function defined on a twisted cotangent bundle over the two-sphere. As a corollary, given any Riemannian two-sphere and a magnetic field on it, there exists a closed magnetic geodesic for almost all kinetic energy levels.

Keywords:

Periodic orbits,
almost existence,
magnetic flows,
capacity bounds,
twisted cotangent bundles over the two-sphere.

Mathematics Subject Classification: Primary: 37J45; Secondary: 53D40.

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References

[1]	A. Abbondandolo, L. Macarini and G. P. Paternain, On the existence of three closed magnetic geodesics for subcritical energies, Comment. Math. Helv., 90 (2015), 155-193.doi: 10.4171/CMH/350.
[2]	A. Abbondandolo, L. Macarini, M. Mazzucchelli and G. P. Paternain, Infinitely many periodic orbits of exact magnetic flows on surfaces for almost every subcritical energy level, preprint, arXiv:1404.7641, (2014).
[3]	V. I. Arnol'd, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.
[4]	L. Asselle and G. Benedetti, Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds, preprint, arXiv:1412.0531, (2014).
[5]	L. Asselle and G. Benedetti, Infinitely many periodic orbits of non-exact oscillating magnetic fields on surfaces with genus at least two for almost every low energy level, to appear in Calc. Var. Partial Differential Equations, 2015.doi: 10.1007/s00526-015-0834-1.
[6]	K. Cieliebak, U. Frauenfelder and G. P. Paternain, Symplectic topology of Mañé's critical values, Geom. Topol., 14 (2010), 1765-1870.doi: 10.2140/gt.2010.14.1765.
[7]	G. Contreras, The Palais-Smale condition on contact type energy levels for convex Lagrangian systems, Calc. Var. Partial Differential Equations, 27 (2006), 321-395.doi: 10.1007/s00526-005-0368-z.
[8]	A. Floer, H. Hofer and D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J., 80 (1995), 251-292.doi: 10.1215/S0012-7094-95-08010-7.
[9]	U. Frauenfelder, V. L. Ginzburg and F. Schlenk, Energy capacity inequalities via an action selector, in Geometry, Spectral Theory, Groups, and Dynamics, Contemp. Math., 387, Amer. Math. Soc., Providence, RI, 2005, 129-152.doi: 10.1090/conm/387/07239.
[10]	U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel J. Math., 159 (2007), 1-56.doi: 10.1007/s11856-007-0037-3.
[11]	V. L. Ginzburg, New generalizations of Poincaré's geometric theorem, Funktsional. Anal. i Prilozhen., 21 (1987), 16-22, 96.
[12]	V. L. Ginzburg and B. Z. Gürel, Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles, Duke Math. J., 123 (2004), 1-47.doi: 10.1215/S0012-7094-04-12311-5.
[13]	H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math., 45 (1992), 583-622.doi: 10.1002/cpa.3160450504.
[14]	H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo, Invent. Math., 90 (1987), 1-9.doi: 10.1007/BF01389030.
[15]	H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994.doi: 10.1007/978-3-0348-8540-9.
[16]	K. Irie, Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits, preprint, arXiv:1112.5247, (2011).
[17]	K. Irie, Hofer-Zehnder capacity of unit disk cotangent bundles and the loop product, J. Eur. Math. Soc. (JEMS), 16 (2014), 2477-2497.doi: 10.4171/JEMS/491.
[18]	F. Lalonde and D. McDuff, $J$-curves and the classification of rational and ruled symplectic $4$-manifolds, in Contact and Symplectic Geometry (Cambridge, 1994), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 3-42.
[19]	G. Liu and G. Tian, Weinstein conjecture and GW-invariants, Commun. Contemp. Math., 2 (2000), 405-459.doi: 10.1142/S0219199700000256.
[20]	G. Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres, Kyushu J. Math., 52 (1998), 331-351.doi: 10.2206/kyushujm.52.331.
[21]	G. Lu, Gromov-Witten invariants and pseudo symplectic capacities, Israel J. Math., 156 (2006), 1-63.doi: 10.1007/BF02773823.
[22]	L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Commun. Contemp. Math., 6 (2004), 913-945.doi: 10.1142/S0219199704001550.
[23]	L. Macarini and F. Schlenk, A refinement of the Hofer-Zehnder theorem on the existence of closed characteristics near a hypersurface, Bull. London Math. Soc., 37 (2005), 297-300.doi: 10.1112/S0024609304003923.
[24]	D. McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc., 3 (1990), 679-712.doi: 10.2307/1990934.
[25]	D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.
[26]	D. McDuff and J. Slimowitz, Hofer-Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol., 5 (2001), 799-830.doi: 10.2140/gt.2001.5.799.
[27]	W. J. Merry, Closed orbits of a charge in a weakly exact magnetic field, Pacific J. Math., 247 (2010), 189-212.doi: 10.2140/pjm.2010.247.189.
[28]	S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.
[29]	S. P. Novikov and I. Shmel'tser, Periodic solutions of Kirchhoff equations for the free motion of a rigid body in a fluid and the extended Lyusternik-Shnirel'man-Morse theory. I, Funktsional. Anal. i Prilozhen., 15 (1981), 54-66.
[30]	L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., Extra Vol. II, 1998, 401-410.
[31]	F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv., 81 (2006), 105-121.doi: 10.4171/CMH/45.
[32]	M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface, Bol. Soc. Brasil. Mat. (N.S.), 20 (1990), 49-58.doi: 10.1007/BF02585433.
[33]	I. A. Taĭmanov, Closed extremals on two-dimensional manifolds, Uspekhi Mat. Nauk, 47 (1992), 143-185, 223.doi: 10.1070/RM1992v047n02ABEH000880.