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Izv. Akad. Nauk SSSR Ser. Mat., 1991, Volume 55, Issue 4, Pages 747–779 (Mi izv987)  

This article is cited in 19 scientific papers (total in 19 papers)

A bordism theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom. A new topological invariant of higher-dimensional integrable systems

A. T. Fomenko

Abstract: Some new objects, bordisms of integrable systems, are found and studied. The classes of rigidly bordant systems form a nontrivial abelian group, which makes it possible to construct new integrable systems on the basis of previously known ones. Among the generators of this bordism group are known physical integrable systems, as, for example, the Lagrange system (from the dynamics of a heavy rigid body) and others. Moreover, a new topological invariant of systems with many degrees of freedom is also constructed. It turns out that two integrable systems are topologically equivalent if and only if their invariants coincide. In particular, it follows from this that the set of topological classes of integrable systems is discrete. The invariant can be effectively calculated for concrete integrable systems arising in physics and mechanics.

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English version:
Mathematics of the USSR-Izvestiya, 1992, 39:1, 731–759

Bibliographic databases:

UDC: 513.944
MSC: Primary 58F05, 57N10, 57M50, 57R90, 57R95; Secondary 55R50, 57N37, 58E15, 70E15, 70H99, 58F07, 55N15
Received: 17.12.1990

Citation: A. T. Fomenko, “A bordism theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom. A new topological invariant of higher-dimensional integrable systems”, Izv. Akad. Nauk SSSR Ser. Mat., 55:4 (1991), 747–779; Math. USSR-Izv., 39:1 (1992), 731–759

Citation in format AMSBIB
\by A.~T.~Fomenko
\paper A~bordism theory for integrable nondegenerate Hamiltonian systems with two degrees of freedom. A~new topological invariant of higher-dimensional integrable systems
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1991
\vol 55
\issue 4
\pages 747--779
\jour Math. USSR-Izv.
\yr 1992
\vol 39
\issue 1
\pages 731--759

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    This publication is cited in the following articles:
    1. L. M. Lerman, Ya. L. Umanskii, “Classification of four-dimensional integrable Hamiltonian systems and Poisson actions of $\mathbb{R}^2$ in extended neighborhoods of simple singular points. I”, Russian Acad. Sci. Sb. Math., 77:2 (1994), 511–542  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. E. N. Selivanova, “A topological portrait of certain multidimensional Hamiltonian systems”, Russian Math. Surveys, 48:2 (1993), 204–205  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. A. Oshemkov, “Classification of hyperbolic singularities of rank zero of integrable Hamiltonian systems”, Sb. Math., 201:8 (2010), 1153–1191  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. V. V. Trofimov, M. V. Shamolin, “Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems”, J. Math. Sci., 180:4 (2012), 365–530  mathnet  crossref  mathscinet
    5. Novikov D.V., “Topologiya izoenergeticheskikh poverkhnostei dlya integriruemogo sluchaya sokolova na algebre li so(3,1)}”, Vestnik Moskovskogo universiteta. Seriya 1: Matematika. Mekhanika, 2011, no. 4, 62–65  elib
    6. M. P. Kharlamov, P. E. Ryabov, “Net diagrams for the Fomenko invariant in the integrable system with three degrees of freedom”, Dokl. Math, 86:3 (2012), 839  crossref
    7. Kharlamov M.P., Ryabov P.E., “Setevye diagrammy dlya invarianta fomenko v integriruemoi sisteme s tremya stepenyami svobody”, Doklady akademii nauk, 447:5 (2012), 499–499  elib
    8. P. E. Ryabov, “Phase topology of one irreducible integrable problem in the dynamics of a rigid body”, Theoret. and Math. Phys., 176:2 (2013), 1000–1015  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. N. S. Slavina, “Topological classification of systems of Kovalevskaya-Yehia type”, Sb. Math., 205:1 (2014), 101–155  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. S. S. Nikolaenko, “A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid”, Sb. Math., 205:2 (2014), 224–268  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. M. P. Kharlamov, P. E. Ryabov, “Topological atlas of the Kovalevskaya top in a double field”, J. Math. Sci., 223:6 (2017), 775–809  mathnet  crossref  mathscinet  elib
    12. I. N. Shnurnikov, “Realizability of singular levels of Morse functions as unions of geodesies”, Moscow University Mathematics Bulletin, 70:6 (2015), 270–273  mathnet  crossref  mathscinet  isi
    13. E. O. Kantonistova, “Topological classification of integrable Hamiltonian systems in a potential field on surfaces of revolution”, Sb. Math., 207:3 (2016), 358–399  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. I. M. Nikonov, “Height atoms whose symmetry groups act transitively on their vertex sets”, Moscow University Mathematics Bulletin, 71:6 (2016), 233–241  mathnet  crossref  mathscinet  isi
    15. D. S. Timonina, “Liouville classification of integrable geodesic flows on a torus of revolution in a potential field”, Moscow University Mathematics Bulletin, 72:3 (2017), 121–128  mathnet  crossref  mathscinet  isi  elib
    16. D. S. Timonina, “Liouville classification of integrable geodesic flows in a potential field on two-dimensional manifolds of revolution: the torus and the Klein bottle”, Sb. Math., 209:11 (2018), 1644–1676  mathnet  crossref  crossref  adsnasa  isi  elib
    17. Fomenko A.T. Vedyushkina V.V., “Singularities of Integrable Liouville Systems, Reduction of Integrals to Lower Degree and Topological Billiards: Recent Results”, Theor. Appl. Mech., 46:1 (2019), 47–63  crossref  isi
    18. A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrability in geometry and physics. New scope and new potential”, Moscow University Mathematics Bulletin, 74:3 (2019), 98–107  mathnet  crossref  mathscinet  isi
    19. E. E. Karginova, “Billiards bounded by arcs of confocal quadrics on the Minkowski plane”, Sb. Math., 211:1 (2020), 1–28  mathnet  crossref  crossref
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