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Uspekhi Mat. Nauk, 1984, Volume 39, Issue 2(236), Pages 3–56 (Mi umn2264)  

This article is cited in 31 scientific papers (total in 31 papers)

Liouville integrability of Hamiltonian systems on Lie algebras

V. V. Trofimov, A. T. Fomenko

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English version:
Russian Mathematical Surveys, 1984, 39:2, 1–67

Bibliographic databases:

UDC: 513.944
MSC: 37K10, 37K30, 37K65, 53D25, 17B20
Received: 04.10.1983

Citation: V. V. Trofimov, A. T. Fomenko, “Liouville integrability of Hamiltonian systems on Lie algebras”, Uspekhi Mat. Nauk, 39:2(236) (1984), 3–56; Russian Math. Surveys, 39:2 (1984), 1–67

Citation in format AMSBIB
\by V.~V.~Trofimov, A.~T.~Fomenko
\paper Liouville integrability of Hamiltonian systems on Lie algebras
\jour Uspekhi Mat. Nauk
\yr 1984
\vol 39
\issue 2(236)
\pages 3--56
\jour Russian Math. Surveys
\yr 1984
\vol 39
\issue 2
\pages 1--67

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  mathscinet  zmath
    2. A. V. Bolsinov, “Involutory families of functions on dual spaces of Lie algebras of type $G\underset\varphi+ V$”, Russian Math. Surveys, 42:6 (1987), 227–228  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. 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
    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. L. M. Tsalenko, “The Laplace–Casimir operators on contractions of the Lie groups $O(n,\mathbb C)$”, Russian Math. Surveys, 44:1 (1989), 261–262  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. 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
    9. V. Zeitlin, “Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure”, Physica D: Nonlinear Phenomena, 49:3 (1991), 353  crossref
    10. Dariusz Chruściński, “Symplectic structure of the von Neumann equation”, Reports on Mathematical Physics, 29:1 (1991), 95  crossref
    11. O. I. Bogoyavlenskii, “Euler equations on finite-dimensional Lie coalgebras, arising in problems of mathematical physics”, Russian Math. Surveys, 47:1 (1992), 117–189  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. T. L. Mordasheva, “Canonical coordinates on orbits of a co-adjoint representation of certain semidirect products of Lie groups”, Russian Math. Surveys, 50:6 (1995), 1282–1283  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    13. T. A. Ivanova, A. D. Popov, “Self-dual Yang–Mills fields in $d=4$ and integrable systems in $1\leq d\leq 3$”, Theoret. and Math. Phys., 102:3 (1995), 280–304  mathnet  crossref  mathscinet  zmath  isi
    14. A. M. Boyarsky, T. V. Skrypnik, “Singular orbits of the co-adjoint representation of Euclidean groups”, Russian Math. Surveys, 55:3 (2000), 564–566  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    15. Antonello Pasini, Vinicio Pelino, “A unified view of Kolmogorov and Lorenz systems”, Physics Letters A, 275:5-6 (2000), 435  crossref  elib
    16. D. V. Georgievskii, M. V. Shamolin, “Valerii Vladimirovich Trofimov”, Journal of Mathematical Sciences, 154:4 (2008), 449–461  mathnet  crossref  mathscinet  zmath
    17. M. M. Zhdanova, “Completely integrable Hamiltonian systems on semidirect sums of Lie algebras”, Sb. Math., 200:5 (2009), 629–659  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. A. S. Vorontsov, “Invariants of Lie algebras representable as semidirect sums with a commutative ideal”, Sb. Math., 200:8 (2009), 1149–1164  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. A. A. Korotkevich, “Integrable Hamiltonian systems on low-dimensional Lie algebras”, Sb. Math., 200:12 (2009), 1731–1766  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    20. T. A. Lepskii, “Incomplete integrable Hamiltonian systems with complex polynomial Hamiltonian of small degree”, Sb. Math., 201:10 (2010), 1511–1538  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. 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
    22. E. A. Kudryavtseva, T. A. Lepskii, “The topology of Lagrangian foliations of integrable systems with hyperelliptic Hamiltonian”, Sb. Math., 202:3 (2011), 373–411  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. Vorontsov A.S., “Kronekerovy indeksy algebry li i otsenka stepenei invariantov”, Vestnik Moskovskogo universiteta. Seriya 1: Matematika. Mekhanika, 2011, no. 1, 26–30  elib
    24. A. M. Izosimov, “Stability of stationary rotations of multidimensional rigid body”, Moscow University Mathematics Bulletin, 68:1 (2013), 80–82  mathnet  crossref  mathscinet
    25. A. Yu. Konyaev, “Classification of Lie algebras with generic orbits of dimension 2 in the coadjoint representation”, Sb. Math., 205:1 (2014), 45–62  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. K. R. Aleshkin, “The topology of integrable systems with incomplete fields”, Sb. Math., 205:9 (2014), 1264–1278  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    27. A. T. Fomenko, A. Yu. Konyaev, “Geometry, dynamics and different types of orbits”, J. Fixed Point Theory Appl, 2014  crossref
    28. T. Šukilović, “Isometry groups of $4$-dimensional nilpotent Lie groups”, J. Math. Sci., 225:4 (2017), 711–721  mathnet  crossref  mathscinet  elib
    29. D. A. Fedoseev, “Bifurcation diagrams of natural Hamiltonian systems on Bertrand manifolds”, Moscow University Mathematics Bulletin, 70:1 (2015), 44–47  mathnet  crossref  mathscinet  isi
    30. E. O. Kantonistova, “Liouville classification of integrable Hamiltonian systems on surfaces of revolution”, Moscow University Mathematics Bulletin, 70:5 (2015), 220–222  mathnet  crossref  mathscinet  isi
    31. A. Yu. Konyaev, “Completeness of commutative Sokolov-Odesskii subalgebras and Nijenhuis operators on $\operatorname{gl}(n)$”, Sb. Math., 211:4 (2020), 583–593  mathnet  crossref  crossref  mathscinet  isi  elib
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