American Institute of Mathematical Sciences

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

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, 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.
Keywords: almost existence, twisted cotangent bundles over the two-sphere., Periodic orbits, capacity bounds, magnetic flows.
Mathematics Subject Classification: Primary: 37J45; Secondary: 53D4.
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), 155-193. 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, arXiv:1404.7641, (2014). Google Scholar

[3]

V. I. Arnol'd, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.  Google Scholar

[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). 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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[11]

V. L. Ginzburg, New generalizations of Poincaré's geometric theorem, Funktsional. Anal. i Prilozhen., 21 (1987), 16-22, 96.  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), 1-47. doi: 10.1215/S0012-7094-04-12311-5.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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K. Irie, Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits, preprint, arXiv:1112.5247, (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), 2477-2497. 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), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 3-42.  Google Scholar

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G. Liu and G. Tian, Weinstein conjecture and GW-invariants, Commun. Contemp. Math., 2 (2000), 405-459. doi: 10.1142/S0219199700000256.  Google Scholar

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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.  Google Scholar

[21]

G. Lu, Gromov-Witten invariants and pseudo symplectic capacities, Israel J. Math., 156 (2006), 1-63. doi: 10.1007/BF02773823.  Google Scholar

[22]

L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Commun. Contemp. Math., 6 (2004), 913-945. 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), 297-300. doi: 10.1112/S0024609304003923.  Google Scholar

[24]

D. McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc., 3 (1990), 679-712. doi: 10.2307/1990934.  Google Scholar

[25]

D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.  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), 54-66.  Google Scholar

[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.  Google Scholar

[31]

F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv., 81 (2006), 105-121. 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), 49-58. doi: 10.1007/BF02585433.  Google Scholar

[33]

I. A. Taĭmanov, Closed extremals on two-dimensional manifolds, Uspekhi Mat. Nauk, 47 (1992), 143-185, 223. 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), 155-193. 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, arXiv:1404.7641, (2014). Google Scholar

[3]

V. I. Arnol'd, Some remarks on flows of line elements and frames, Dokl. Akad. Nauk SSSR, 138 (1961), 255-257.  Google Scholar

[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). 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), 1765-1870. 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), 321-395. 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), 251-292. 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, Spectral Theory, Groups, and Dynamics, Contemp. Math., 387, Amer. Math. Soc., Providence, RI, 2005, 129-152. 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), 1-56. 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), 16-22, 96.  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), 1-47. 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), 583-622. 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), 1-9. 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], Birkhäuser Verlag, Basel, 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, arXiv:1112.5247, (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), 2477-2497. 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), Publ. Newton Inst., 8, Cambridge Univ. Press, Cambridge, 1996, 3-42.  Google Scholar

[19]

G. Liu and G. Tian, Weinstein conjecture and GW-invariants, Commun. Contemp. Math., 2 (2000), 405-459. 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), 331-351. doi: 10.2206/kyushujm.52.331.  Google Scholar

[21]

G. Lu, Gromov-Witten invariants and pseudo symplectic capacities, Israel J. Math., 156 (2006), 1-63. doi: 10.1007/BF02773823.  Google Scholar

[22]

L. Macarini, Hofer-Zehnder capacity and Hamiltonian circle actions, Commun. Contemp. Math., 6 (2004), 913-945. 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), 297-300. doi: 10.1112/S0024609304003923.  Google Scholar

[24]

D. McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc., 3 (1990), 679-712. doi: 10.2307/1990934.  Google Scholar

[25]

D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology, Amer. Math. Soc. Colloq. Publ., 52, American Mathematical Society, Providence, RI, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk, 37 (1982), 3-49, 248.  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), 54-66.  Google Scholar

[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.  Google Scholar

[31]

F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv., 81 (2006), 105-121. 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), 49-58. doi: 10.1007/BF02585433.  Google Scholar

[33]

I. A. Taĭmanov, Closed extremals on two-dimensional manifolds, Uspekhi Mat. Nauk, 47 (1992), 143-185, 223. doi: 10.1070/RM1992v047n02ABEH000880.  Google Scholar

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