# American Institute of Mathematical Sciences

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

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

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

Received  March 2015 Published  June 2015

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.
Citation: Gabriele Benedetti, Kai Zehmisch. On the existence of periodic orbits for magnetic systems on the two-sphere. Journal of Modern Dynamics, 2015, 9: 141-146. doi: 10.3934/jmd.2015.9.141
##### 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. doi: 10.4171/CMH/350. Google Scholar [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, (2014). Google Scholar [3] V. I. Arnol'd, Some remarks on flows of line elements and frames,, Dokl. Akad. Nauk SSSR, 138 (1961), 255. Google Scholar [4] L. Asselle and G. Benedetti, Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds,, preprint, (2014). Google Scholar [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. Google Scholar [6] K. Cieliebak, U. Frauenfelder and G. P. Paternain, Symplectic topology of Mañé's critical values,, Geom. Topol., 14 (2010), 1765. doi: 10.2140/gt.2010.14.1765. Google Scholar [7] G. Contreras, The Palais-Smale condition on contact type energy levels for convex Lagrangian systems,, Calc. Var. Partial Differential Equations, 27 (2006), 321. doi: 10.1007/s00526-005-0368-z. Google Scholar [8] A. Floer, H. Hofer and D. Salamon, Transversality in elliptic Morse theory for the symplectic action,, Duke Math. J., 80 (1995), 251. doi: 10.1215/S0012-7094-95-08010-7. Google Scholar [9] U. Frauenfelder, V. L. Ginzburg and F. Schlenk, Energy capacity inequalities via an action selector,, in Geometry, (2005), 129. doi: 10.1090/conm/387/07239. Google Scholar [10] U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds,, Israel J. Math., 159 (2007), 1. doi: 10.1007/s11856-007-0037-3. Google Scholar [11] V. L. Ginzburg, New generalizations of Poincaré's geometric theorem,, Funktsional. Anal. i Prilozhen., 21 (1987), 16. Google Scholar [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. doi: 10.1215/S0012-7094-04-12311-5. Google Scholar [13] H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres,, Comm. Pure Appl. Math., 45 (1992), 583. doi: 10.1002/cpa.3160450504. Google Scholar [14] H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo,, Invent. Math., 90 (1987), 1. doi: 10.1007/BF01389030. Google Scholar [15] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], (1994). doi: 10.1007/978-3-0348-8540-9. Google Scholar [16] K. Irie, Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits,, preprint, (2011). Google Scholar [17] K. Irie, Hofer-Zehnder capacity of unit disk cotangent bundles and the loop product,, J. Eur. Math. Soc. (JEMS), 16 (2014), 2477. doi: 10.4171/JEMS/491. Google Scholar [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), 3. Google Scholar [19] G. Liu and G. Tian, Weinstein conjecture and GW-invariants,, Commun. Contemp. Math., 2 (2000), 405. doi: 10.1142/S0219199700000256. Google Scholar [20] G. Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres,, Kyushu J. Math., 52 (1998), 331. doi: 10.2206/kyushujm.52.331. Google Scholar [21] G. Lu, Gromov-Witten invariants and pseudo symplectic capacities,, Israel J. Math., 156 (2006), 1. doi: 10.1007/BF02773823. Google Scholar [22] L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions,, Commun. Contemp. Math., 6 (2004), 913. doi: 10.1142/S0219199704001550. Google Scholar [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. doi: 10.1112/S0024609304003923. Google Scholar [24] D. McDuff, The structure of rational and ruled symplectic $4$-manifolds,, J. Amer. Math. Soc., 3 (1990), 679. doi: 10.2307/1990934. Google Scholar [25] D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology,, Amer. Math. Soc. Colloq. Publ., (2004). Google Scholar [26] D. McDuff and J. Slimowitz, Hofer-Zehnder capacity and length minimizing Hamiltonian paths,, Geom. Topol., 5 (2001), 799. doi: 10.2140/gt.2001.5.799. Google Scholar [27] W. J. Merry, Closed orbits of a charge in a weakly exact magnetic field,, Pacific J. Math., 247 (2010), 189. doi: 10.2140/pjm.2010.247.189. Google Scholar [28] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory,, Uspekhi Mat. Nauk, 37 (1982), 3. Google Scholar [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. Google Scholar [30] L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms,, in Proceedings of the International Congress of Mathematicians, (1998), 401. Google Scholar [31] F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics,, Comment. Math. Helv., 81 (2006), 105. doi: 10.4171/CMH/45. Google Scholar [32] M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface,, Bol. Soc. Brasil. Mat. (N.S.), 20 (1990), 49. doi: 10.1007/BF02585433. Google Scholar [33] I. A. Taĭmanov, Closed extremals on two-dimensional manifolds,, Uspekhi Mat. Nauk, 47 (1992), 143. doi: 10.1070/RM1992v047n02ABEH000880. Google Scholar

show all references

##### 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. doi: 10.4171/CMH/350. Google Scholar [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, (2014). Google Scholar [3] V. I. Arnol'd, Some remarks on flows of line elements and frames,, Dokl. Akad. Nauk SSSR, 138 (1961), 255. Google Scholar [4] L. Asselle and G. Benedetti, Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds,, preprint, (2014). Google Scholar [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. Google Scholar [6] K. Cieliebak, U. Frauenfelder and G. P. Paternain, Symplectic topology of Mañé's critical values,, Geom. Topol., 14 (2010), 1765. doi: 10.2140/gt.2010.14.1765. Google Scholar [7] G. Contreras, The Palais-Smale condition on contact type energy levels for convex Lagrangian systems,, Calc. Var. Partial Differential Equations, 27 (2006), 321. doi: 10.1007/s00526-005-0368-z. Google Scholar [8] A. Floer, H. Hofer and D. Salamon, Transversality in elliptic Morse theory for the symplectic action,, Duke Math. J., 80 (1995), 251. doi: 10.1215/S0012-7094-95-08010-7. Google Scholar [9] U. Frauenfelder, V. L. Ginzburg and F. Schlenk, Energy capacity inequalities via an action selector,, in Geometry, (2005), 129. doi: 10.1090/conm/387/07239. Google Scholar [10] U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds,, Israel J. Math., 159 (2007), 1. doi: 10.1007/s11856-007-0037-3. Google Scholar [11] V. L. Ginzburg, New generalizations of Poincaré's geometric theorem,, Funktsional. Anal. i Prilozhen., 21 (1987), 16. Google Scholar [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. doi: 10.1215/S0012-7094-04-12311-5. Google Scholar [13] H. Hofer and C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres,, Comm. Pure Appl. Math., 45 (1992), 583. doi: 10.1002/cpa.3160450504. Google Scholar [14] H. Hofer and E. Zehnder, Periodic solutions on hypersurfaces and a result by C. Viterbo,, Invent. Math., 90 (1987), 1. doi: 10.1007/BF01389030. Google Scholar [15] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics,, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], (1994). doi: 10.1007/978-3-0348-8540-9. Google Scholar [16] K. Irie, Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits,, preprint, (2011). Google Scholar [17] K. Irie, Hofer-Zehnder capacity of unit disk cotangent bundles and the loop product,, J. Eur. Math. Soc. (JEMS), 16 (2014), 2477. doi: 10.4171/JEMS/491. Google Scholar [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), 3. Google Scholar [19] G. Liu and G. Tian, Weinstein conjecture and GW-invariants,, Commun. Contemp. Math., 2 (2000), 405. doi: 10.1142/S0219199700000256. Google Scholar [20] G. Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres,, Kyushu J. Math., 52 (1998), 331. doi: 10.2206/kyushujm.52.331. Google Scholar [21] G. Lu, Gromov-Witten invariants and pseudo symplectic capacities,, Israel J. Math., 156 (2006), 1. doi: 10.1007/BF02773823. Google Scholar [22] L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions,, Commun. Contemp. Math., 6 (2004), 913. doi: 10.1142/S0219199704001550. Google Scholar [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. doi: 10.1112/S0024609304003923. Google Scholar [24] D. McDuff, The structure of rational and ruled symplectic $4$-manifolds,, J. Amer. Math. Soc., 3 (1990), 679. doi: 10.2307/1990934. Google Scholar [25] D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology,, Amer. Math. Soc. Colloq. Publ., (2004). Google Scholar [26] D. McDuff and J. Slimowitz, Hofer-Zehnder capacity and length minimizing Hamiltonian paths,, Geom. Topol., 5 (2001), 799. doi: 10.2140/gt.2001.5.799. Google Scholar [27] W. J. Merry, Closed orbits of a charge in a weakly exact magnetic field,, Pacific J. Math., 247 (2010), 189. doi: 10.2140/pjm.2010.247.189. Google Scholar [28] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory,, Uspekhi Mat. Nauk, 37 (1982), 3. Google Scholar [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. Google Scholar [30] L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms,, in Proceedings of the International Congress of Mathematicians, (1998), 401. Google Scholar [31] F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics,, Comment. Math. Helv., 81 (2006), 105. doi: 10.4171/CMH/45. Google Scholar [32] M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface,, Bol. Soc. Brasil. Mat. (N.S.), 20 (1990), 49. doi: 10.1007/BF02585433. Google Scholar [33] I. A. Taĭmanov, Closed extremals on two-dimensional manifolds,, Uspekhi Mat. Nauk, 47 (1992), 143. doi: 10.1070/RM1992v047n02ABEH000880. Google Scholar
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