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Sbornik: Mathematics, 2015, Volume 206, Issue 5, Pages 718–737
DOI: https://doi.org/10.1070/SM2015v206n05ABEH004476 (Mi sm8425) 		 
This article is cited in 7 scientific papers (total in 7 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
English version PDF (536 kB) Citations (7)
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DOI: https://doi.org/10.1070/SM2015v206n05ABEH004476
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.
Funding agency Grant number
Ministry of Education and Science of the Russian Federation НШ-1410.2012.1
Russian Foundation for Basic Research 13-01-00664-а
This research was supported by the Programme of the President of the Russian Federation for Support of Leading Scientific Schools (grant no. НШ-1410.2012.1) and by the Russian Foundation for Basic Research (grant no. 13-01-00664-a).
Received: 25.09.2014
Russian version:
Matematicheskii Sbornik, 2015, Volume 206, Number 5, Pages 107–126
DOI: https://doi.org/10.4213/sm8425
Bibliographic databases:
Document Type: Article
UDC: 514.853
MSC: Primary 70F17; Secondary 53A20, 53A35, 70G45, 70H12
Language: English
Original paper language: Russian
Citation: E. A. Kudryavtseva, D. A. Fedoseev, “Mechanical systems with closed orbits on manifolds of revolution”, Mat. Sb., 206:5 (2015), 107–126; Sb. Math., 206:5 (2015), 718–737
Citation in format AMSBIB
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    • This publication is cited in the following 7 articles:
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    Russian version PDF:153
    English version PDF:8
    References:41
    First page:19
     
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