Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds

S. S. Nikolaenko

Sbornik. Mathematics, Tome 211 (2020) no. 8, pp. 1127-1158

Résumé

We construct a complete topological invariant of foliations of finite type defined by smooth functions on two-dimensional noncompact orientable manifolds. In particular, we describe a complete topological classification of noncompact bifurcations of such foliations. We establish a natural one-to-one correspondence between the set of all such bifurcations and the set of oriented coloured graphs of a special form. As a consequence, we obtain the Liouville and trajectory classifications of Hamiltonian systems of finite type on noncompact two-dimensional manifolds. Bibliography: 25 titles.

Détail

Keywords: Hamiltonian system, topological classification, trajectory equivalence, noncompact atom, $f$-graph.

Bibliographie

[1] A. V. Bolsinov, S. V. Matveev, A. T. Fomenko, “Topological classification of integrable Hamiltonian systems with two degrees of freedom. List of systems of small complexity”, Russian Math. Surveys, 45:2 (1990), 59–94 | DOI | MR | Zbl

[2] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl

[3] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[4] A. T. Fomenko, “Topological invariants of Liouville integrable Hamiltonian systems”, Funct. Anal. Appl., 22:4 (1988), 286–296 | DOI | MR | Zbl

[5] A. T. Fomenko, H. Zieschang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Math. USSR-Izv., 36:3 (1991), 567–596 | DOI | MR | Zbl

[6] A. V. Bolsinov, “A smooth trajectory classification of integrable Hamiltonian systems with two degrees of freedom”, Sb. Math., 186:1 (1995), 1–27 | DOI | MR | Zbl

[7] A. V. Bolsinov, A. T. Fomenko, “Traektornaya ekvivalentnost integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody. Teorema klassifikatsii. I”, Matem. sb., 185:4 (1994), 27–80 ; II:5, 27–78 ; A. V. Bolsinov, A. T. Fomenko, “Orbital equivalence of integrable Hamiltonian systems with two degrees of freedom. A classification theorem. I”, Russian Acad. Sci. Sb. Math., 81:2 (1995), 421–465 ; II, 82:1, 21–63 | MR | Zbl | MR | Zbl | DOI | DOI

[8] A. V. Bolsinov, A. T. Fomenko, “Orbital invariants of integrable Hamiltonian systems. The case of simple systems. Orbital classification of systems of Euler type in rigid body dynamics”, Izv. Math., 59:1 (1995), 63–100 | DOI | MR | Zbl

[9] A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | DOI | MR | MR | Zbl | Zbl

[10] V. V. Vedyushkina, A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733 | DOI | DOI | MR | Zbl

[11] V. V. Fokicheva, “Classification of billiard motions in domains bounded by confocal parabolas”, Sb. Math., 205:8 (2014), 1201–1221 | DOI | DOI | MR | Zbl

[12] O. A. Zagryadskii, “Bertranovskie sistemy i ikh fazovoe prostranstvo”, Nauka i obrazovanie, 2014, no. 12, 365–386

[13] D. A. Fedoseev, “Bifurcation diagrams of natural Hamiltonian systems on Bertrand manifolds”, Moscow Univ. Math. Bull., 70:1 (2015), 44–47 | DOI | MR | Zbl

[14] D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$”, Sb. Math., 202:5 (2011), 749–781 | DOI | DOI | MR | Zbl

[15] S. S. Nikolaenko, “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

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

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

[18] E. A. Kudryavtseva, T. A. Lepskii, “Integriruemye gamiltonovy sistemy s nepolnymi potokami i mnogougolniki Nyutona”, Sovremennye problemy matematiki i mekhaniki, 6:3 (2011), 42–55

[19] E. A. Kudryavtseva, T. A. Lepskii, “The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian”, Sb. Math., 202:3 (2011), 373–411 | DOI | DOI | MR | Zbl

[20] K. R. Aleshkin, “The topology of integrable systems with incomplete fields”, Sb. Math., 205:9 (2014), 1264–1278 | DOI | DOI | MR | Zbl

[21] E. A. Kudryavtseva, “An analogue of the Liouville theorem for integrable Hamiltonian systems with incomplete flows”, Dokl. Math., 86:1 (2012), 527–529 | DOI | MR | Zbl

[22] E. A. Kudryavtseva, T. A. Lepskii, “Topologiya sloenii i teorema Liuvillya dlya integriruemykh sistem s nepolnymi potokami”, Tr. sem. po vektornomu i tenzornomu analizu, no. 27, 2011, 106–149

[23] V. V. Sharko, “Gladkie funktsii na nekompaktnykh poverkhnostyakh”, Zb. prats In-tu matem. NAN Ukraïni, 3:3 (2006), 443–473 | Zbl

[24] A. O. Prishlyak, “Topological equivalence of smooth functions with isolated critical points on a closed surface”, Topology Appl., 119:3 (2002), 257–267 | DOI | MR | Zbl

[25] A. A. Oshemkov, “Morse functions on two-dimensional surfaces. Encoding of singularities”, Proc. Steklov Inst. Math., 205 (1995), 119–127 | MR | Zbl