Superintegrable Bertrand Natural Mechanical Systems

Kudryavtseva, Elena; Fedoseev, D. A.

doi:10.1007/s10958-020-04882-2

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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. Natural mechanical systems of Bertrand's type were described under several restrictions by Bertrand [1], Darboux [2], Besse [3], Perlick [4], Zagryadskii, Kudryavtseva and Fedoseev [5], Kudryavtseva and Fedoseev [6,7,8] and others (cf. a survey in [5]).…”

Section: Introductionmentioning

confidence: 99%

Superintegrable Bertrand magnetic geodesic flows

Kudryavtseva,

Podlipaev

2020

Preprint

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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.

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