Topological classification of non-compact 3-atoms with a circle action

S. S. Nikolaenko

Čebyševskij sbornik, Tome 22 (2021) no. 5, pp. 185-197.

Résumé

For integrable Hamiltonian systems with two degrees of freedom we investigate the topology of the Liouville foliation in a 3-dimensional non-compact invariant neighborhood of a singular leaf. All the singularities of the system are supposed to be non-degenerate. In the case when all the leaves of the Liouville foliation are compact, this problem is already solved: the well-known A. T. Fomenko theorem states that any non-degenerate 3-dimensional singularity (3-atom) is an $S^1$-fibration of the special type (Seifert fibration) over a 2-dimensional singularity (2-atom). Thus, the problem of the topological classification of 3-atoms is reduced to the significantly more simple classification problem for 2-atoms (i. e. singularities of foliations determined by Morse functions on 2-surfaces). The latter problem is well-studied in the framework of the Fomenko classification theory for integrable systems. In the non-compact case, the set of all 3-atoms becomes much richer. That is why we consider only 3-atoms satisfying the following conditions: completeness of the Hamiltonian flows generated by the first integrals of the system, finiteness of the number of orbits of the Hamiltonian $\mathbb{R}^2$-action on the singular leaf, and existence among these orbits of a non-contractible one. Under these restrictions, we proof that the 3-atom admits a Hamiltonian locally free $S^1$-action preserving the leaves of the Liouville foliation. As a corollary, we obtain the analogue of the Fomenko theorem and thus reduce the classification problem for non-compact 3-atoms satisfying the above conditions to the similar classification problem for non-compact 2-atoms that we solved earlier.

Détail

Keywords: integrable Hamiltonian system, non-compact atom, circle action, Seifert fibration, Hamiltonian action.

Bibliographie

[1] Bolsinov A. V., Fomenko A. T., Integriruemye gamiltonovy sistemy. Geometriya, topologiya, klassifikatsiya, V 2 t., Izd. dom “Udmurtskii universitet”, Izhevsk, 1999, 444 pp. | MR

[2] Fedoseev D. A., Fomenko A. T., “Nekompaktnye osobennosti integriruemykh dinamicheskikh sistem”, Fundament. i prikl. matem., 21:6 (2016), 217–243

[3] Novikov D. V., “Topologicheskie osobennosti integriruemogo sluchaya Sokolova na algebre Li e(3)”, Matem. sb., 202:5 (2011), 127–160 | DOI | MR

[4] Nikolaenko S. S., “Topological classification of the Goryachev Integrable systems in the rigid body dynamics: non-compact case”, Lobachevskii J. Math., 38:6 (2017), 1050–1060 | DOI | MR | Zbl

[5] Borisov A. V., Mamaev I. S., “Rigid Body Dynamics in Non-Euclidean Spaces”, Russ. J. Math. Phys., 23:4 (2016), 431–454 | DOI | MR | Zbl

[6] Kibkalo V. A., “Svoistvo nekompaktnosti sloev i osobennostei neevklidovoi sistemy Kovalevskoi na puchke algebr Li”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 6, 56–59 | MR | Zbl

[7] Vedyushkina (Fokicheva) V. V., Fomenko A. T., “Integriruemye topologicheskie billiardy i ekvivalentnye dinamicheskie sistemy”, Izv. RAN. Ser. matem., 81:4 (2017), 20–67 | DOI | MR | Zbl

[8] Fiorani E., Giachetta G., Sardanashvily G., “The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds”, J. Phys. A, 36:7 (2003), 101–107 | DOI | MR

[9] Kudryavtseva E. A., “Analog teoremy Liuvillya dlya integriruemykh gamiltonovykh sistem s nepolnymi potokami”, Dokl. RAN, 445:4 (2012), 383–385 | MR | Zbl

[10] Kudryavtseva E. A., Lepskii T. A., “Topologiya sloenii i teorema Liuvillya dlya integriruemykh sistem s nepolnymi potokami”, Tr. sem. po vektornomu i tenzornomu analizu, 27, 2011, 106–149

[11] Nikolaenko S. S., “Topologicheskaya klassifikatsiya gamiltonovykh sistem na dvumernykh nekompaktnykh mnogoobraziyakh”, Matem. sb., 211:8 (2020), 68–101 | DOI | MR | Zbl

[12] Fomenko A. T., “Topologicheskie invarianty gamiltonovykh sistem, integriruemykh po Liuvillyu”, Funkts. analiz i ego pril., 22:4 (1988), 38–51 | MR | Zbl

[13] Fomenko A. T., “Teoriya Morsa integriruemykh gamiltonovykh sistem”, Dokl. AN SSSR, 287:5 (1986), 1071–1075 | MR | Zbl

[14] Fomenko A. T., “Topologiya poverkhnostei postoyannoi energii nekotorykh integriruemykh gamiltonovykh sistem i prepyatstviya k integriruemosti”, Izv. AN SSSR. Ser. matem., 50:6 (1986), 1276–1307 | MR | Zbl

[15] Kudryavtseva E. A., “Hidden toric symmetry and structural stability of singularities in integrable systems”, European Journal of Mathematics, 2021, 63 pp. (published 25 October 2021) | DOI | MR

[16] Bau T., Zung N. T., “Singularities of integrable and near-integrable Hamiltonian systems”, J. Nonlinear Sci., 7:1 (1997), 1–7 | DOI | MR | Zbl

[17] Zung N. T., “A note on degenerate corank-one singularities of integrable Hamiltonian systems”, Comment. Math. Helv., 75:2 (2000), 271–283 | DOI | MR | Zbl

[18] Kudryavtseva E. A., Martynchuk N. N., “Existence of a smooth Hamiltonian circle action near parabolic orbits and cuspidal tori”, Regular and Chaotic Dynamics, 26:6 (2021), 732–741 | DOI | MR | Zbl

[19] Matveev S. V., Fomenko A. T., Algoritmicheskie i kompyuternye metody v trekhmernoi topologii, izd-vo MGU, M., 1991, 304 pp. | MR