The topology of integral submanifolds of completely integrable Hamiltonian systems

A. V. Brailov   ; A. T. Fomenko

Sbornik. Mathematics, Tome 62 (1989) no. 2, pp. 373-383

Résumé

It is proved that the class $(X)$ of three-dimensional closed compact manifolds that are constant energy surfaces of integrable (by means of a Bott integral) Hamiltonian systems coincides precisely with the class $(Q)$ of three-dimensional orientable manifolds admitting decomposition into “circular handles”. Fomenko previously proved the inclusion $(X)\subset(Q)$. An explicit geometric description is also given for modifications of Liouville tori in neighborhoods of nonorientable critical submanifolds of the moment mapping of an integrable system. Figures: 1. Bibliography: 20 titles.

Bibliographie

[1] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR

[2] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii. Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982 | MR

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

[4] Smale S., “Topology and mechanics”, Invent. Math., 10:6 (1970), 305–331 ; 11:1, 45–64 | DOI | MR | Zbl | DOI | MR | Zbl

[5] Kharlamov M. P., “Topologicheskii analiz klassicheskikh integriruemykh sistem v dinamike tverdogo tela”, DAN SSSR, 273:6 (1983), 1322–1325 | MR | Zbl

[6] Kharlamov M. P., “K issledovaniyu oblastei vozmozhnosti dvizheniya v mekhanicheskikh sistemakh”, DAN SSSR, 267:3 (1982), 571–573 | MR | Zbl

[7] Pogosyan T. I., Kharlamov M. P., “Oblasti vozmozhnosti dvizheniya v nekotorykh mekhanicheskikh sistemakh”, PMM, 45:4 (1981), 605–610 | MR

[8] Tatarinov Ya. V., “Portrety klassicheskikh integralov zadachi o vraschenii tverdogo tela vokrug nepodvizhnoi tochki”, Vestn. MGU. Ser. matem., mekh., 1974, no. 6, 99–105 | MR | Zbl

[9] Kozlov V. V., Metody kachestvennogo analiza v dinamike tverdogo tela, Izd-vo Mosk. un-ta, M., 1980 | MR | Zbl

[10] Bogoyavlenskii O. I., Ivakh G. F., “Topologicheskii analiz integriruemykh sluchaev B. A. Steklova”, UMN, 40:4 (1985), 145–146 | MR | Zbl

[11] Iakob A., “Invariant manifolds in the motion of a rigid body about a fixed point”, Rev. Roum. Pure et Appl., 16:10 (1971), 1497–1521 | MR

[12] Lacomba E. A., “Invariant sets and polhodes in the rigid body problem”, Celestial Mechanics., 21:1 (1980), 45–53 | DOI | MR | Zbl

[13] Waldhausen F., “Eine klasse von 3-dimensional mannigfaltigkeiten I”, Invent. Math., 3:4 (1967), 308–333 | DOI | MR | Zbl

[14] Fomenko A. T., “Topologiya trekhmernykh mnogoobrazii i integriruemye mekhanicheskie gamiltonovy sistemy”, V Tiraspolskii simpoz. po obschei topologii i ee prilozheniyam, Kishinev, 1985, 235–237

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

[16] Trofimov V. V., Fomenko A. T., “Geometriya skobok Puassona i metody integrirovaniya po Liuvillyu sistem na simmetricheskikh prostranstvakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 29, VINITI, M., 1986, 3–108 | MR

[17] Fomenko A. T., “Nekotorye novye topologicheskie zadachi gamiltonovoi mekhaniki”, Geometriya, differentsialnye uravneniya i mekhanika, Izd-vo MGU, M., 1986, 19–24 | MR

[18] Matveev S. V., Burmistrova A. B., “Struktura $S$-funktsii na orientiruemykh 3-mnogoobraziyakh”, Kh1 Vsesoyuz. shkola po teorii operatorov v funktsion. prostranstvakh (Chelyabinsk 1986 g.), Chelyabinsk, 1986, 12

[19] Oshemkov A. A., “Bottovskie integraly nekotorykh integriruemykh gamiltonovykh sistem”, Geometriya, differentsialnye uravneniya i mekhanika, Izd-vo MGU, M., 1986, 115–117 | MR

[20] Fomenko A. T., Zieschang H., On the topology of three-dimensional manifolds arising in Hamiltonian mechanics, Preprint. Bericht Nr. 55/1985., Ruhr – Universität Bochum, 1985 | MR