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

This article is cited in 12 scientific papers (total in 12 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.
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English version:
Sbornik: Mathematics, 2012, 203:8, 1112–1150

Bibliographic databases:

UDC: 514.853
MSC: Primary 70F17; Secondary 53A20, 53A35, 70B05, 70H06, 70H12, 70H33
Received: 29.03.2011 and 31.03.2012

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
\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
\jour Sb. Math.
\yr 2012
\vol 203
\issue 8
\pages 1112--1150

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    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  mathnet  crossref  mathscinet
    2. A. T. Fomenko, A. Yu. Konyaev, “Geometry, dynamics and different types of orbits”, J. Fixed Point Theory Appl., 15:1 (2014), 49–66  crossref  mathscinet  zmath  isi  elib  scopus
    3. O. A. Zagryadskii, “The relations between the Bertrand, Bonnet, and Tannery classes”, Moscow University Mathematics Bulletin, 69:6 (2014), 277–279  mathnet  crossref  mathscinet
    4. E. A. Kudryavtseva, D. A. Fedoseev, “Mechanical systems with closed orbits on manifolds of revolution”, Sb. Math., 206:5 (2015), 718–737  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. D. A. Fedoseev, “Bifurcation diagrams of natural Hamiltonian systems on Bertrand manifolds”, Moscow University Mathematics Bulletin, 70:1 (2015), 44–47  mathnet  crossref  mathscinet
    7. O. A. Zagryadskii, “Bertrand surfaces with a pseudo-Riemannian metric of revolution”, Moscow University Mathematics Bulletin, 70:1 (2015), 49–52  mathnet  crossref  mathscinet
    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  mathnet  crossref  mathscinet
    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  mathnet  crossref  mathscinet  zmath  elib
    10. E. A. Kudryavtseva, D. A. Fedoseev, “The Bertrand's manifolds with equators”, Moscow University Mathematics Bulletin, 71:1 (2016), 23–26  mathnet  crossref  mathscinet  isi
    11. D. A. Fedoseev, A. T. Fomenko, “Nekompaktnye osobennosti integriruemykh dinamicheskikh sistem”, Fundament. i prikl. matem., 21:6 (2016), 217–243  mathnet
    12. Martynchuk N. Dullin H.R. Efstathiou K. Waalkens H., “Scattering Invariants in Euler'S Two-Center Problem”, Nonlinearity, 32:4 (2019), 1296–1326  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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