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

Previous Article
Ergodicity and topological entropy of geodesic flows on surfaces
JMD Home
This Volume
Next Article
The relative cohomology of abelian covers of the flat pillowcase

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

Abstract
Related Papers

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.
[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.

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.
[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.

[1]	César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617
[2]	Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005
[3]	Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61
[4]	Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205
[5]	Bryce Weaver. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. Journal of Modern Dynamics, 2014, 8 (2) : 139-176. doi: 10.3934/jmd.2014.8.139
[6]	Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[7]	Ping-Liang Huang, Youde Wang. Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1775-1795. doi: 10.3934/dcdss.2016074
[8]	Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[9]	Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595
[10]	Radu Pantilie. On the embeddings of the Riemann sphere with nonnegative normal bundles. Electronic Research Announcements, 2018, 25: 87-95. doi: 10.3934/era.2018.25.009
[11]	Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291
[12]	Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158
[13]	C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935
[14]	Krzysztof Frączek, Mariusz Lemańczyk. A class of mixing special flows over two--dimensional rotations. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4823-4829. doi: 10.3934/dcds.2015.35.4823
[15]	Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315-321. doi: 10.3934/proc.2009.2009.315
[16]	Peng Huang. Existence of invariant curves for degenerate almost periodic reversible mappings. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022074
[17]	Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357
[18]	Regina Martínez. On the existence of doubly symmetric "Schubart-like" periodic orbits. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 943-975. doi: 10.3934/dcdsb.2012.17.943
[19]	Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171
[20]	Luis García-Naranjo. Reduction of almost Poisson brackets and Hamiltonization of the Chaplygin sphere. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 37-60. doi: 10.3934/dcdss.2010.3.37

2021 Impact Factor: 0.641

Tools

Metrics

Other articles
by authors

on AIMS
Gabriele Benedetti Kai Zehmisch
on Google Scholar
Gabriele Benedetti Kai Zehmisch

[Back to Top]