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.
Citation:
O. A. Zagryadskii, E. A. Kudryavtseva, D. A. Fedoseev, “A generalization of Bertrand's theorem to surfaces of revolution”, Sb. Math., 203:8 (2012), 1112–1150
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\paper A generalization of Bertrand's theorem to surfaces of revolution
\jour Sb. Math.
\yr 2012
\vol 203
\issue 8
\pages 1112--1150
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Linking options:
https://www.mathnet.ru/eng/sm7870
https://doi.org/10.1070/SM2012v203n08ABEH004257
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Ortega R., Rojas D., “A Proof of Bertrand'S Theorem Using the Theory of Isochronous Potentials”, J. Dyn. Differ. Equ., 31:4 (2019), 2017–2028
E. A. Kudryavtseva, S. A. Podlipaev, “Superintegrable Bertrand magnetic geodesic flows”, J. Math. Sci., 259:5 (2021), 689–698
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E. A. Kudryavtseva, D. A. Fedoseev, “The Bertrand's manifolds with equators”, Moscow University Mathematics Bulletin, 71:1 (2016), 23–26
D. A. Fedoseev, A. T. Fomenko, “Noncompact bifurcations of integrable dynamic systems”, J. Math. Sci., 248:6 (2020), 810–827
E. A. Kudryavtseva, D. A. Fedoseev, “Mechanical systems with closed orbits on manifolds of revolution”, Sb. Math., 206:5 (2015), 718–737
I. V. Sypchenko, D. S. Timonina, “Closed geodesics on piecewise smooth surfaces of revolution with constant curvature”, Sb. Math., 206:5 (2015), 738–769
D. A. Fedoseev, “Bifurcation diagrams of natural Hamiltonian systems on Bertrand manifolds”, Moscow University Mathematics Bulletin, 70:1 (2015), 44–47
O. A. Zagryadskii, “Bertrand surfaces with a pseudo-Riemannian metric of revolution”, Moscow University Mathematics Bulletin, 70:1 (2015), 49–52
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
Richard Montgomery, Corey Shanbrom, Fields Institute Communications, 73, Geometry, Mechanics, and Dynamics, 2015, 319
A. T. Fomenko, A. Yu. Konyaev, “Geometry, dynamics and different types of orbits”, J. Fixed Point Theory Appl., 15:1 (2014), 49–66
O. A. Zagryadskii, “The relations between the Bertrand, Bonnet, and Tannery classes”, Moscow University Mathematics Bulletin, 69:6 (2014), 277–279
Anatoly T. Fomenko, Andrei Konyaev, Solid Mechanics and Its Applications, 211, Continuous and Distributed Systems, 2014, 3
O. A. Zagryadskii, D. A. Fedoseev, “The explicit form of the Bertrand metric”, Moscow University Mathematics Bulletin, 68:5 (2013), 258–262
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