# 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:

- 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}

}

Other availability

Save to My Library

You must Sign In or Create an Account in order to save documents to your library.