skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: A generalization of Bertrand's theorem to surfaces of revolution

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.

Authors:
; ;  [1]
  1. M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
22094064
Resource Type:
Journal Article
Journal Name:
Sbornik. Mathematics
Additional Journal Information:
Journal Volume: 203; Journal Issue: 8; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ADDITIVES; CENTRAL POTENTIAL; CLASSIFICATION; EQUATOR; SURFACES

Citation Formats

Zagryadskii, Oleg A, Kudryavtseva, Elena A, and Fedoseev, Denis A. A generalization of Bertrand's theorem to surfaces of revolution. United States: N. p., 2012. Web. doi:10.1070/SM2012V203N08ABEH004257.
Zagryadskii, Oleg A, Kudryavtseva, Elena A, & Fedoseev, Denis A. A generalization of Bertrand's theorem to surfaces of revolution. United States. https://doi.org/10.1070/SM2012V203N08ABEH004257
Zagryadskii, Oleg A, Kudryavtseva, Elena A, and Fedoseev, Denis A. 2012. "A generalization of Bertrand's theorem to surfaces of revolution". United States. https://doi.org/10.1070/SM2012V203N08ABEH004257.
@article{osti_22094064,
title = {A generalization of Bertrand's theorem to surfaces of revolution},
author = {Zagryadskii, Oleg A and Kudryavtseva, Elena A and Fedoseev, Denis A},
abstractNote = {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.},
doi = {10.1070/SM2012V203N08ABEH004257},
url = {https://www.osti.gov/biblio/22094064}, journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 8,
volume = 203,
place = {United States},
year = {Fri Aug 31 00:00:00 EDT 2012},
month = {Fri Aug 31 00:00:00 EDT 2012}
}