This article is cited in 9 papers

Mechanical systems with closed orbits on manifolds of revolution

E. A. Kudryavtseva, D. A. Fedoseev

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study natural mechanical systems describing the motion of a particle on a two-dimensional Riemannian manifold of revolution in the field of a central smooth potential. We obtain a classification of Riemannian manifolds of revolution and central potentials on them that have the strong Bertrand property: any nonsingular (that is, not contained in a meridian) orbit is closed. We also obtain a classification of manifolds of revolution and central potentials on them that have the ‘stable’ Bertrand property: every parallel is an ‘almost stable’ circular orbit, and any nonsingular bounded orbit is closed.
Bibliography: 14 titles.

Keywords: Bertrand Riemannian manifold, surface of revolution, equator, Tannery manifold, Maupertuis' principle.

UDC: 514.853

MSC: Primary 70F17; Secondary 53A20, 53A35, 70G45, 70H12

Received: 25.09.2014

DOI: 10.4213/sm8425

 English version: Sbornik: Mathematics, 2015, 206:5, 718–737

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