Journal of Modern Dynamics (JMD)

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

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

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Gabriele Benedetti - Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany (email)
Kai Zehmisch - Mathematisches Institut, Westfälische Wilhelms- Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany (email)

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.

Received: March 2015;      Available Online: June 2015.