Geodesic
Mechanical systems with closed orbits on manifolds of revolution
Sbornik. Mathematics, Tome 206 (2015) no. 5, pp. 718-737

Voir la notice de l'article provenant de la source Math-Net.Ru

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
Mots-clés : Maupertuis' principle. 
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E. A. Kudryavtseva; D. A. Fedoseev. Mechanical systems with closed orbits on manifolds of revolution. Sbornik. Mathematics, Tome 206 (2015) no. 5, pp. 718-737. http://geodesic.mathdoc.fr/item/SM_2015_206_5_a4/