Geodesic
Superintegrable Bertrand magnetic geodesic flows
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 6, pp. 169-182

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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 metric has a constant scalar curvature and the magnetic field is homogeneous, i.e., proportional to the area form. 
@article{FPM_2019_22_6_a6,
     author = {E. A. Kudryavtseva and S. A. Podlipaev},
     title = {Superintegrable {Bertrand} magnetic geodesic flows},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {169--182},
     publisher = {mathdoc},
     volume = {22},
     number = {6},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2019_22_6_a6/}
}
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E. A. Kudryavtseva; S. A. Podlipaev. Superintegrable Bertrand magnetic geodesic flows. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 6, pp. 169-182. http://geodesic.mathdoc.fr/item/FPM_2019_22_6_a6/