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

Gabriele  Benedetti; Kai  Zehmisch

doi:10.3934/jmd.2015.9.141

Article Contents

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

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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

Received: February 28, 2015

Published: May 2015

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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.

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