Mechanical systems with closed orbits on manifolds of revolution

E. A. Kudryavtseva   ; D. A. Fedoseev

Sbornik. Mathematics, Tome 206 (2015) no. 5, pp. 718-737

Résumé

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.

Détail

Keywords: Bertrand Riemannian manifold, surface of revolution, equator, Tannery manifold
Mots-clés : Maupertuis' principle.

Bibliographie

[1] Nguyen Tien Zung, A. T. Fomenko, “Topological classification of integrable non-degenerate Hamiltonians on a constant energy three-dimensional sphere”, Russian Math. Surveys, 45:6 (1990), 109–135 | DOI | MR | Zbl

[2] A. V. Bolsinov, A. T. Fomenko, “Integrable geodesic flows on the sphere, generated by Goryachev–Chaplygin and Kowalewski systems in the dynamics of a rigid body”, Math. Notes, 56:2 (1994), 859–861 | DOI | MR | Zbl

[3] A. V. Bolsinov, A. T. Fomenko, “Application of classification theory for integrable Hamiltonian systems to geodesic flows on 2-sphere and 2-torus and to the description of the topological structure of momentum mapping near singular point”, J. Math. Sci., 78:5 (1996), 542–555 | DOI | MR | Zbl

[4] A. V. Bolsinov, V. S. Matveev, A. T. Fomenko, “Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry”, Sb. Math., 189:10 (1998), 1441–1466 | DOI | DOI | MR | Zbl

[5] 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 | DOI | DOI | MR | Zbl

[6] A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., 93, Springer-Verlag, Berlin–New York, 1978, ix+262 pp. | MR | MR | Zbl

[7] M. Santoprete, “Gravitational and harmonic oscillator potentials on surfaces of revolution”, J. Math. Phys., 49:4 (2008), 042903, 16 pp. | DOI | MR | Zbl

[8] J. Bertrand, “Théorème relatif au mouvement d'un point attiré vers un centre fixe”, C. R. Math. Acad. Sci. Paris, 77 (1873), 849–853 ; F. C. Santos, V. Soares, A. C. Tort, An English translation of Bertrand's theorem, 2007, 6 pp., arXiv: 0704.2396 | Zbl

[9] V. Perlick, “Bertrand spacetimes”, Classical Quantum Gravity, 9:4 (1992), 1009–1021 | DOI | MR | Zbl

[10] O. A. Zagryadskii, D. A. Fedoseev, “O globalnoi i lokalnoi realizuemosti rimanovykh mnogoobrazii Bertrana v vide poverkhnostei vrascheniya”, Vestn. Mosk. un-ta. Ser. 1 Matem. Mekh., 2015, no. 3 (to appear)

[11] D. A. Fedoseev, O. A. Zagryadskii, “The explicit form of the Bertrand metric”, Moscow Univ. Math. Bull., 68:5 (2013), 258–262 | DOI | MR

[12] E. A. Kudryavtseva, D. A. Fedoseev, “O mnogoobraziyakh Bertrana s ekvatorami”, Vestn. Mosk. un-ta. Ser. 1 Matem. Mekh., 2016 (to appear)

[13] A. V. Bolsinov, V. V. Kozlov, A. T. Fomenko, “The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body”, Russian Math. Surveys, 50:3 (1995), 473–501 | DOI | MR | Zbl

[14] G. Darboux, Leçons sur la théorie générale des surfaces et les allpications géométriques du calcul infinitésimal. Troisième partie, Réimpression de la première édition de 1894, Chelsea Publishing Co., Bronx, N. Y., 1972, xvi+501 pp. | MR | Zbl