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.

[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).

[3]

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

[4]

L. Asselle and G. Benedetti, Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds,, preprint, (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. 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. 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. 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, (2005), 129. doi: 10.1090/conm/387/07239.

[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.

[11]

V. L. Ginzburg, New generalizations of Poincaré's geometric theorem,, Funktsional. Anal. i Prilozhen., 21 (1987), 16.

[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.

[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.

[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.

[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.

[16]

K. Irie, Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits,, preprint, (2011).

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[24]

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

[25]

D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology,, Amer. Math. Soc. Colloq. Publ., (2004).

[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.

[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.

[28]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory,, Uspekhi Mat. Nauk, 37 (1982), 3.

[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.

[30]

L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms,, in Proceedings of the International Congress of Mathematicians, (1998), 401.

[31]

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

[33]

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

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.

[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).

[3]

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

[4]

L. Asselle and G. Benedetti, Periodic orbits of magnetic flows for weakly exact unbounded forms and for spherical manifolds,, preprint, (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. 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. 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. 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, (2005), 129. doi: 10.1090/conm/387/07239.

[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.

[11]

V. L. Ginzburg, New generalizations of Poincaré's geometric theorem,, Funktsional. Anal. i Prilozhen., 21 (1987), 16.

[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.

[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.

[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.

[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.

[16]

K. Irie, Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits,, preprint, (2011).

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[24]

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

[25]

D. McDuff and D. Salamon, $J$-holomorphic Curves and Symplectic Topology,, Amer. Math. Soc. Colloq. Publ., (2004).

[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.

[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.

[28]

S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory,, Uspekhi Mat. Nauk, 37 (1982), 3.

[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.

[30]

L. Polterovich, Geometry on the group of Hamiltonian diffeomorphisms,, in Proceedings of the International Congress of Mathematicians, (1998), 401.

[31]

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

[33]

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

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