A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom

A. T. Fomenko ; H. Zieschang

Izvestiya. Mathematics , Tome 36 (1991) no. 3, pp. 567-596.

Résumé

A new topological invariant is constructed which classifies integrable Hamiltonian systems with two degrees of freedom (admitting a Bott integral). A criterion for the equivalence of Bott systems is proved: such systems are topologically equivalent if and only if their topological invariants coincide. The topological invariant is effectively calculated for specific integrable Hamiltonian systems in physics and mechanics.

Bibliographie

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

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

[3] Fomenko A. T., Tsishang X., “O topologii trekhmernykh mnogoobrazii, voznikayuschikh v gamiltonovoi mekhanike”, DAN SSSR, 294:2 (1987), 283–287 | MR | Zbl

[4] Fomenko A. T., Tsishang X., “O tipichnykh topologicheskikh svoistvakh integriruemykh gamiltonovykh sistem”, Izv. AN SSSR. Ser. matem., 52:2 (1988), 378–407 | Zbl

[5] Fomenko A. T., “New topological invariant of integrable Hamiltonians”, Baku International Topological Conference, Abstracts. Part 2, Matem. in-t AN SSSR, In-t matematiki i mekhaniki AN AzerbSSR, Baku, 1987, 316

[6] Trofimov V. V., Fomenko A. T., “Geometricheskie i algebraicheskie mekhanizmy integriruemosti gamiltonovykh sistem na odnorodnykh prostranstvakh i algebrakh Li”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 16, VINITI, M., 1987, 227–299 | MR

[7] Matveev S. V., Fomenko A. T., Zharko V. V., Kruglye funktsii Morsa i izoenergeticheskie poverkhnosti integriruemykh gamiltonovykh sistem, Preprint No 86, 76, In-t matematiki AN USSR, Kiev, 1986, 32 pp.

[8] Waldhausen F., “Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten, I, II”, Invent. Math., 3:4 (1967), 308–333 | DOI | MR | Zbl

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

[10] Brailov A. V., Fomenko A. T., “Topologiya integralnykh podmnogoobrazii vpolne integriruemykh gamiltonovykh sistem”, Matem. sb., 134(176):3 (1987), 375–385 | MR | Zbl

[11] Kozlov V. V., “Integriruemost i neintegriruemost v gamiltonovoi mekhanike”, UMN, 38:1 (1983), 3–67 | MR | Zbl

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

[13] Novikov S. P., “Variatsionnye metody i periodicheskie resheniya uravnenii tipa Kirkhgofa, II”, Funkts. analiz i ego prilozh., 15:4 (1981), 37–52 | MR

[14] Novikov S. P., “Gamiltonov formalizm i mnogoznachnyi analog teorii Morsa”, UMN, 37:5 (1982), 3–49 | MR | Zbl

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

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

[17] Fomenko A. T., Zieschang H., On typical properties of integrable Hamiltonian system, Preprint, No 92/1987, Univ. Bochum, Germany, 1–47 | MR

[18] Matveev S. V., Fomenko A. T., “Izoenergeticheskie poverkhnosti gamiltonovykh sistem, perechislenie trekhmernykh mnogoobrazii v poryadke vozrastaniya ikh slozhnosti i vychislenie ob'emov zamknutykh giperbolicheskikh mnogoobrazii”, UMN, 43:1 (1988), 5–22 | MR | Zbl

[19] Matveev S. V., Fomenko A. T., “Teoriya tipa Morsa dlya integriruemykh gamiltonovykh sistem s ruchnymi integralami”, Matem. zametki, 43:5 (1988), 663–671 | MR

[20] Matveev S. V., Fomenko A. T., Sharko V. V., “Kruglye funktsii Morsa i izoenergeticheskie poverkhnosti integriruemykh gamiltonovykh sistem”, Matem. sb., 135(177):3 (1968), 325–346

[21] Oshemkov A. A., “Topologiya izoenergeticheskikh poverkhnostei i bifurkatsionnye diagrammy integriruemykh sluchaev dinamiki tverdogo tela na $so(4)$”, UMN, 42:6 (1987), 199–200 | MR

[22] Zieschang H., Vogt E., Coldwey H. D., Surfaces and planar discontinuous groups, Lect. Notes in Math., 835, Springer-Verlag, 1980 | MR | Zbl

[23] Orlik P., Vogt E., Zieschang H., “Zur Topologie gefaserter dreidimensionaler Mannigfaltigkeiten”, Topology, 6:1 (1967), 49–65 | DOI | MR

[24] Oshemkov A. A., “The phase topology of some integrable Hamiltonian systems on $so(4)$”, Bakinskaya mezhdunarodnaya topologicheskaya konferentsiya, Ch. 2. Tezisy, Matem. in-t AN SSSR, In-t matematiki i mekhaniki AN AzerbSSR, Baku, 1987, 230

[25] Lyndon R., Schupp P., Combinatorial group theory, Springer-Verlag, Berlin, Heidelberg, New York, 1977 | MR | Zbl

[26] Channell P. J., Scovel C., Symplectfc integration of Hamiltonian systems, Preprint, LA-UR-88-1828, Los Alamos National Laboratory, Los Alamos, USA, 1988

[27] Kharlamov M. P., Topologicheskii analiz integriruemykh zadach dinamiki tverdogo tela, LGU, L., 1988 | MR

[28] Johannson K., Homotopy equivalences of $3$-manifolds with boundaries, 761, Springer LNM, 1979 | MR | Zbl

[29] Jaco W., Shalen P. B., “Surface homeomorphisms and periodicity”, Topology, 16:4 (1977), 347–367 | DOI | MR | Zbl