RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Sb.: Year: Volume: Issue: Page: Find

 Mat. Sb., 2012, Volume 203, Number 8, Pages 39–78 (Mi msb7870)

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.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm7870

Full text: PDF file (944 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 514.853
MSC: Primary 70F17; Secondary 53A20, 53A35, 70B05, 70H06, 70H12, 70H33

Citation: O. A. Zagryadskii, E. A. Kudryavtseva, D. A. Fedoseev, “A generalization of Bertrand's theorem to surfaces of revolution”, Mat. Sb., 203:8 (2012), 39–78; Sb. Math., 203:8 (2012), 1112–1150

Citation in format AMSBIB
\Bibitem{ZagKudFed12} \by O.~A.~Zagryadskii, E.~A.~Kudryavtseva, D.~A.~Fedoseev \paper A generalization of Bertrand's theorem to surfaces of revolution \jour Mat. Sb. \yr 2012 \vol 203 \issue 8 \pages 39--78 \mathnet{http://mi.mathnet.ru/msb7870} \crossref{https://doi.org/10.4213/sm7870} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3024812} \zmath{https://zbmath.org/?q=an:06110267} \elib{http://elibrary.ru/item.asp?id=19066553} \transl \jour Sb. Math. \yr 2012 \vol 203 \issue 8 \pages 1112--1150 \crossref{https://doi.org/10.1070/SM2012v203n08ABEH004257} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309818600003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84868621826} 

• http://mi.mathnet.ru/eng/msb7870
• https://doi.org/10.4213/sm7870
• http://mi.mathnet.ru/eng/msb/v203/i8/p39

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. A. Zagryadskii, D. A. Fedoseev, “The explicit form of the Bertrand metric”, Moscow University Mathematics Bulletin, 68:5 (2013), 258–262
2. A. T. Fomenko, A. Yu. Konyaev, “Geometry, dynamics and different types of orbits”, J. Fixed Point Theory Appl., 15:1 (2014), 49–66
3. O. A. Zagryadskii, “The relations between the Bertrand, Bonnet, and Tannery classes”, Moscow University Mathematics Bulletin, 69:6 (2014), 277–279
4. E. A. Kudryavtseva, D. A. Fedoseev, “Mechanical systems with closed orbits on manifolds of revolution”, Sb. Math., 206:5 (2015), 718–737
5. I. V. Sypchenko, D. S. Timonina, “Closed geodesics on piecewise smooth surfaces of revolution with constant curvature”, Sb. Math., 206:5 (2015), 738–769
6. D. A. Fedoseev, “Bifurcation diagrams of natural Hamiltonian systems on Bertrand manifolds”, Moscow University Mathematics Bulletin, 70:1 (2015), 44–47
7. O. A. Zagryadskii, “Bertrand surfaces with a pseudo-Riemannian metric of revolution”, Moscow University Mathematics Bulletin, 70:1 (2015), 49–52
8. O. A. Zagryadskii, D. A. Fedoseev, “The global and local realizability of Bertrand Riemannian manifolds as surfaces of revolution”, Moscow University Mathematics Bulletin, 70:3 (2015), 119–124
9. Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Spatial Problem of 2 Bodies on a Sphere. Reduction and Stochasticity”, Regul. Chaotic Dyn., 21:5 (2016), 556–580
10. E. A. Kudryavtseva, D. A. Fedoseev, “The Bertrand's manifolds with equators”, Moscow University Mathematics Bulletin, 71:1 (2016), 23–26
11. D. A. Fedoseev, A. T. Fomenko, “Nekompaktnye osobennosti integriruemykh dinamicheskikh sistem”, Fundament. i prikl. matem., 21:6 (2016), 217–243
12. Martynchuk N. Dullin H.R. Efstathiou K. Waalkens H., “Scattering Invariants in Euler'S Two-Center Problem”, Nonlinearity, 32:4 (2019), 1296–1326
•  Number of views: This page: 433 Full text: 133 References: 48 First page: 34