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Title: Mechanical systems with closed orbits on manifolds of revolution

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.

Authors:
;  [1]
  1. M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
22590456
Resource Type:
Journal Article
Journal Name:
Sbornik. Mathematics
Additional Journal Information:
Journal Volume: 206; Journal Issue: 5; Other Information: Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; MATHEMATICAL MANIFOLDS; RIEMANN SPACE; TWO-DIMENSIONAL CALCULATIONS

Citation Formats

Kudryavtseva, E A, and Fedoseev, D A. Mechanical systems with closed orbits on manifolds of revolution. United States: N. p., 2015. Web. doi:10.1070/SM2015V206N05ABEH004476.
Kudryavtseva, E A, & Fedoseev, D A. Mechanical systems with closed orbits on manifolds of revolution. United States. doi:10.1070/SM2015V206N05ABEH004476.
Kudryavtseva, E A, and Fedoseev, D A. Sun . "Mechanical systems with closed orbits on manifolds of revolution". United States. doi:10.1070/SM2015V206N05ABEH004476.
@article{osti_22590456,
title = {Mechanical systems with closed orbits on manifolds of revolution},
author = {Kudryavtseva, E A and Fedoseev, D A},
abstractNote = {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.},
doi = {10.1070/SM2015V206N05ABEH004476},
journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 5,
volume = 206,
place = {United States},
year = {2015},
month = {5}
}