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Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 6, Pages 1276–1307 (Mi izv1573)  

This article is cited in 62 scientific papers (total in 62 papers)

The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability

A. T. Fomenko


Abstract: The surfaces of constant energy in integrable Hamiltonian systems which possess Bott integrals are classified. A complete topological classification is given of surgery of Liouville tori in general position in integrable Hamiltonian systems.
Bibliography: 28 titles.

Full text: PDF file (4608 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1987, 29:3, 629–658

Bibliographic databases:

UDC: 513.944
MSC: Primary 58F05, 58F07, 57R65; Secondary 70H10
Received: 14.01.1985

Citation: A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Izv. Akad. Nauk SSSR Ser. Mat., 50:6 (1986), 1276–1307; Math. USSR-Izv., 29:3 (1987), 629–658

Citation in format AMSBIB
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\by A.~T.~Fomenko
\paper The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1986
\vol 50
\issue 6
\pages 1276--1307
\mathnet{http://mi.mathnet.ru/izv1573}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=883162}
\zmath{https://zbmath.org/?q=an:0649.58019|0619.58023}
\transl
\jour Math. USSR-Izv.
\yr 1987
\vol 29
\issue 3
\pages 629--658
\crossref{https://doi.org/10.1070/IM1987v029n03ABEH000986}


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    This publication is cited in the following articles:
    1. A. V. Brailov, A. T. Fomenko, “The topology of integral submanifolds of completely integrable Hamiltonian systems”, Math. USSR-Sb., 62:2 (1989), 373–383  mathnet  crossref  mathscinet  zmath
    2. A. T. Fomenko, H. Zieschang, “On typical topological properties of integrable Hamiltonian systems”, Math. USSR-Izv., 32:2 (1989), 385–412  mathnet  crossref  mathscinet  zmath
    3. S. V. Matveev, A. T. Fomenko, “Constant energy surfaces of Hamiltonian systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds”, Russian Math. Surveys, 43:1 (1988), 3–24  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. S. V. Matveev, A. T. Fomenko, V. V. Sharko, “Round Morse functions and isoenergy surfaces of integrable Hamiltonian systems”, Math. USSR-Sb., 63:2 (1989), 319–336  mathnet  crossref  mathscinet  zmath
    5. A. T. Fomenko, “Topological invariants of Liouville integrable Hamiltonian systems”, Funct. Anal. Appl., 22:4 (1988), 286–296  mathnet  crossref  mathscinet  zmath  isi
    6. A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    7. 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  mathnet  crossref  mathscinet  zmath  adsnasa
    8. 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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. Nguyen Tien Zung, A. T. Fomenko, “Topological classification of integrable non-degenerate Hamiltonians on a constant energy three-dimensional sphere”, Russian Math. Surveys, 45:6 (1990), 109–135  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. Karlheinz Geist, Werner Lauterborn, “The nonlinear dynamics of the damped and driven Toda chain”, Physica D: Nonlinear Phenomena, 41:1 (1990), 1  crossref
    11. A. T. Fomenko, “A topological invariant which roughly classifies integrable strictly nondegenerate Hamiltonians on four-dimensional symplectic manifolds”, Funct. Anal. Appl., 25:4 (1991), 262–272  mathnet  crossref  mathscinet  zmath  isi
    12. A. V. Bolsinov, “Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution”, Math. USSR-Izv., 38:1 (1992), 69–90  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. E. A. Lacomba, J. Llibre, “Integrals, invariant manifolds, and degeneracy for central force problems in Rn”, J Math Phys (N Y ), 33:6 (1992), 2138  crossref  mathscinet  zmath  adsnasa
    14. 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
    15. E. N. Selivanova, “Classification of geodesic flows of Liouville metrics on the two-dimensional torus up to topological equivalence”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 491–505  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. 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. II”, Russian Acad. Sci. Sb. Math., 78:2 (1994), 479–506  mathnet  crossref  mathscinet  zmath  isi
    17. Nguyen Tien Zung, L. S. Polyakova, E. N. Selivanova, “Topological Classification of Integrable Geodesic Flows on Orientable Two-Dimensional Riemannian Manifolds with Additional Integral Depending on Momenta Linearly or Quadratically”, Funct. Anal. Appl., 27:3 (1993), 186–196  mathnet  crossref  mathscinet  zmath  isi
    18. V. V. Kalashnikov, “On genericity of integrable Hamiltonian systems of Bott type”, Russian Acad. Sci. Sb. Math., 81:1 (1995), 87–99  mathnet  crossref  mathscinet  zmath  isi
    19. 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  mathnet  crossref  mathscinet  zmath  isi
    20. A. V. Bolsinov, A. T. Fomenko, “Orbital equivalence of integrable Hamiltonian systems with two degrees of freedom. A classification theorem. II”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 21–63  mathnet  crossref  mathscinet  zmath  isi
    21. E. N. Selivanova, “Orbital isomorphisms of Liouville systems on a two-dimensional torus”, Sb. Math., 186:10 (1995), 1531–1549  mathnet  crossref  mathscinet  zmath  isi
    22. A. V. Bolsinov, A. T. Fomenko, “Orbital Classification of Geodesic Flows on Two-Dimensional Ellipsoids. The Jacobi Problem is Orbitally Equivalent to the Integrable Euler Case in Rigid Body Dynamics”, Funct. Anal. Appl., 29:3 (1995), 149–160  mathnet  crossref  mathscinet  zmath  isi
    23. 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  mathnet  crossref  mathscinet  zmath  isi
    24. K. N. Mishachev, “Hamiltonian links in three-dimensional manifolds”, Izv. Math., 59:6 (1995), 1193–1205  mathnet  crossref  mathscinet  zmath  isi
    25. V. S. Matveev, “Integrable Hamiltonian system with two degrees of freedom. The topological structure of saturated neighbourhoods of points of focus-focus and saddle-saddle type”, Sb. Math., 187:4 (1996), 495–524  mathnet  crossref  crossref  mathscinet  zmath  isi
    26. A. V. Bolsinov, “Fomenko invariants in the theory of integrable Hamiltonian systems”, Russian Math. Surveys, 52:5 (1997), 997–1015  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    27. A. A. Oshemkov, V. V. Sharko, “Classification of Morse–Smale flows on two-dimensional manifolds”, Sb. Math., 189:8 (1998), 1205–1250  mathnet  crossref  crossref  mathscinet  zmath  isi
    28. A. V. Bolsinov, V. S. Matveev, “Singularities of momentum maps of integrable Hamiltonian systems with two degrees of freedom”, Journal of Mathematical Sciences (New York), 94:4 (1999), 1477  crossref  mathscinet
    29. D. Repovš, A. B. Skopenkov, “Obstructions for Seifert fibrations and an extension of the Bolsinov–Fomenko theorem on integrable Hamiltonian systems”, Russian Math. Surveys, 54:3 (1999), 652–653  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    30. A. V. Bolsinov, A. T. Fomenko, “Exact topological classification of Hamiltonian flows on smooth two-dimensional surfaces”, J Math Sci, 94:4 (1999), 1457  mathnet  crossref
    31. A. Yu. Moskvin, “Topology of the Liouville foliation on a 2-sphere in the Dullin-Matveev integrable case”, Sb. Math., 199:3 (2008), 411–448  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    32. E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko, “Maximally symmetric cell decompositions of surfaces and their coverings”, Sb. Math., 199:9 (2008), 1263–1353  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    33. G. Haghighatdoost, A. A. Oshemkov, “The topology of Liouville foliation for the Sokolov integrable case on the Lie algebra so(4)”, Sb. Math., 200:6 (2009), 899–921  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    34. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and stability of integrable systems”, Russian Math. Surveys, 65:2 (2010), 259–318  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    35. V. O. Manturov, “Parity in knot theory”, Sb. Math., 201:5 (2010), 693–733  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    36. 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
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    38. D. V. Novikov, “Topological features of the Sokolov integrable case on the Lie algebra $\mathrm{e}(3)$”, Sb. Math., 202:5 (2011), 749–781  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    39. Izosimov A.M., “Gladkie invarianty osobennostei tipa fokus-fokus”, Vestnik moskovskogo universiteta. seriya 1: matematika. mekhanika, 2011, no. 4, 59–61  elib
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    41. P. E. Ryabov, M. P. Kharlamov, “Classification of singularities in the problem of motion of the Kovalevskaya top in a double force field”, Sb. Math., 203:2 (2012), 257–287  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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    43. Ratyu T., Filatova T.A., Shafarevich A.I., “Nekompaktnye lagranzhevy mnogoobraziya, sootvetstvuyuschie spektralnym seriyam operatora shredingera s delta-potentsialom na poverkhnosti vrascheniya”, Doklady akademii nauk, 446:6 (2012), 618–618  elib
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    45. N. V. Volchanetskii, I. M. Nikonov, “Maximally symmetric height atoms”, Moscow University Mathematics Bulletin, 68:2 (2013), 83–86  mathnet  crossref  mathscinet  elib
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    49. O. A. Zagryadskii, “The relations between the Bertrand, Bonnet, and Tannery classes”, Moscow University Mathematics Bulletin, 69:6 (2014), 277–279  mathnet  crossref  mathscinet
    50. Rasoul Akbarzadeh, Ghorbanali Haghighatdoost, “The Topology of Liouville Foliation for the BorisovMamaevSokolov Integrable Case on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 20:3 (2015), 317–344  mathnet  crossref  mathscinet  zmath  adsnasa
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    54. V. A. Kibkalo, “The topology of the analog of Kovalevskaya integrability case on the Lie algebra $\mathrm{so}(4)$ under zero area integral”, Moscow University Mathematics Bulletin, 71:3 (2016), 119–123  mathnet  crossref  mathscinet  isi
    55. M. A. Tuzhilin, “Singularities of integrable Hamiltonian systems with the same boundary foliation. An infinite series”, Moscow University Mathematics Bulletin, 71:5 (2016), 185–190  mathnet  crossref  mathscinet  isi
    56. 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
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