This article is cited in 20 papers

A generalization of Bertrand's theorem to surfaces of revolution

O. A. Zagryadskii, E. A. Kudryavtseva, D. A. Fedoseev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove a generalization of Bertrand's theorem to the case of abstract surfaces of revolution that have no ‘equators’. We prove a criterion for exactly two central potentials to exist on this type of surface (up to an additive and a multiplicative constant) for which all bounded orbits are closed and there is a bounded nonsingular noncircular orbit. We prove a criterion for the existence of exactly one such potential. We study the geometry and classification of the corresponding surfaces with the aforementioned pair of potentials (gravitational and oscillatory) or unique potential (oscillatory). We show that potentials of the required form do not exist on surfaces that do not belong to any of the classes described.
Bibliography: 33 titles.

Keywords: Bertrand's theorem, inverse problem of dynamics, surface of revolution, motion in a central field, closed orbits.

UDC: 514.853

MSC: Primary 70F17; Secondary 53A20, 53A35, 70B05, 70H06, 70H12, 70H33

Received: 29.03.2011 and 31.03.2012

DOI: 10.4213/sm7870

 English version: Sbornik: Mathematics, 2012, 203:8, 1112–1150

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