Superintegrable Bertrand Magnetic Geodesic Flows

  title={Superintegrable Bertrand Magnetic Geodesic Flows},
  author={Elena A. Kudryavtseva and S. A. Podlipaev},
  journal={Journal of Mathematical Sciences},
  pages={689 - 698}
The problem of description of superintegrable systems (i.e., systems with closed trajectories in a certain domain) in the class of rotationally symmetric natural mechanical systems goes back to Bertrand and Darboux. We describe all superintegrable (in a domain of slow motions) systems in the class of rotationally symmetric magnetic geodesic flows. We show that all sufficiently slow motions in a central magnetic field on a two-dimensional manifold of revolution are periodic if and only if the… 

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  • 2019

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  • 2015