The Multiplicity Problem for Periodic Orbits of Magnetic Flows on the 2-Sphere

Alberto Abbondandolo, Luca Asselle, Gabriele Benedetti, Marco Mazzucchelli, Iskander A. Taimanov

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

We consider magnetic Tonelli Hamiltonian systems on the cotangent bundle of the 2-sphere, where the magnetic form is not necessarily exact. It is known that, on very low and on high energy levels, these systems may have only finitely many periodic orbits. Our main result asserts that almost all energy levels in a precisely characterized intermediate range (e(0), e(1)) possess infinitely many periodic orbits. Such a range of energies is non-empty, for instance, in the physically relevant case where the Tonelli Lagrangian is a kinetic energy and the magnetic form is oscillating (in which case, e(0) = 0 is the minimal energy of the system).

Original languageEnglish
Pages (from-to)17-30
Number of pages14
JournalAdvanced Nonlinear Studies
Volume17
Issue number1
DOIs
Publication statusPublished - 2 Jan 2017

Keywords

  • Tonelli Lagrangians
  • Magnetic Flows
  • Hamiltonian Systems
  • Periodic Orbits
  • Mane Critical Values
  • LAGRANGIAN SYSTEMS
  • GEODESICS
  • SURFACES
  • FIELDS
  • Mañé Critical Values

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