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Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 5, Pages 883–938 (Mi izv1502)  

This article is cited in 32 scientific papers (total in 32 papers)

Integrable Euler equations on Lie algebras arising in problems of mathematical physics

O. I. Bogoyavlenskii

Abstract: Complete integrability in the sense of Liouville is established for the rotation of an arbitrary rigid body about a fixed point in a Newtonian field with an arbitrary homogeneous quadratic potential. Explicit formulas, which express the angular velocity of the rigid body rotation in terms of theta functions on Riemannian surfaces, are obtained. A series of cases is found in which the Euler equations on the Lie algebra $\operatorname{SO}(4)$ are integrable. A model of pulsar rotation, the dynamics of which are described by Euler equations on the Lie algebra $\operatorname{SO}(3)\oplus E_3$, is investigated.
Bibliography: 53 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1985, 25:2, 207–257

Bibliographic databases:

UDC: 517.91
MSC: Primary 58F07; Secondary 58F05, 70E99, 76W05, 83F05, 82A45, 22E70, 34C35, 3
Received: 29.03.1984

Citation: O. I. Bogoyavlenskii, “Integrable Euler equations on Lie algebras arising in problems of mathematical physics”, Izv. Akad. Nauk SSSR Ser. Mat., 48:5 (1984), 883–938; Math. USSR-Izv., 25:2 (1985), 207–257

Citation in format AMSBIB
\by O.~I.~Bogoyavlenskii
\paper Integrable Euler equations on Lie algebras arising in problems of mathematical physics
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1984
\vol 48
\issue 5
\pages 883--938
\jour Math. USSR-Izv.
\yr 1985
\vol 25
\issue 2
\pages 207--257

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    This publication is cited in the following articles:
    1. O. I. Bogoyavlenskii, “Integrable cases of the dynamics of a rigid body, and integrable systems on the spheres $S^n$”, Math. USSR-Izv., 27:2 (1986), 203–218  mathnet  crossref  mathscinet  zmath
    2. O. I. Bogoyavlenskii, “Some constructions of integrable dynamical systems”, Math. USSR-Izv., 31:1 (1988), 47–75  mathnet  crossref  mathscinet  zmath
    3. O. I. Bogoyavlenskii, “Algebraic constructions of certain integrable equations”, Math. USSR-Izv., 33:1 (1989), 39–65  mathnet  crossref  mathscinet  zmath
    4. A. G. Reiman, M. A. Semenov-Tian-Shansky, “Lax representation with a spectral parameter for the kowalevski top and its generalizations”, Funct. Anal. Appl., 22:2 (1988), 158–160  mathnet  crossref  mathscinet  isi
    5. O. I. Bogoyavlenskii, “Breaking solitons in $2+1$-dimensional integrable equations”, Russian Math. Surveys, 45:4 (1990), 1–89  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. A. I. Zhivkov, “Geometry of invariant manifolds of a gyroscope in the field of a quadratic potential”, Math. USSR-Izv., 37:1 (1991), 227–242  mathnet  crossref  mathscinet  zmath  adsnasa
    7. O. I. Bogoyavlenskii, “A theorem on two commuting automorphisms, and integrable differential equations”, Math. USSR-Izv., 36:2 (1991), 263–279  mathnet  crossref  mathscinet  zmath  adsnasa
    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. O. I. Bogoyavlenskii, “Algebraic constructions of integrable dynamical systems-extensions of the Volterra system”, Russian Math. Surveys, 46:3 (1991), 1–64  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. 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
    11. O. I. Bogoyavlenskii, “Integrable problems of the dynamics of coupled rigid bodies”, Russian Acad. Sci. Izv. Math., 41:3 (1993), 395–416  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    12. Philip Holmes, Jeffrey Jenkins, Naomi Ehrich Leonard, “Dynamics of the Kirchhoff equations I: Coincident centers of gravity and buoyancy”, Physica D: Nonlinear Phenomena, 118:3-4 (1998), 311  crossref
    13. Antonello Pasini, Vinicio Pelino, “A unified view of Kolmogorov and Lorenz systems”, Physics Letters A, 275:5-6 (2000), 435  crossref  elib
    14. Yuri B Suris, “Integrable Discretizations of Some Cases of the Rigid Body Dynamics”, Journal of Nonlinear Mathematical Physics, 8:4 (2001), 534  crossref
    15. A. V. Bolsinov, A. V. Borisov, “Compatible Poisson Brackets on Lie Algebras”, Math. Notes, 72:1 (2002), 10–30  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. D. B. Zot'ev, “Symplectic Geometry of Manifolds with Almost Nowhere Vanishing Closed 2-Form”, Math. Notes, 76:1 (2004), 62–72  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    17. D. B. Zot'ev, “Phase topology of Appelrot class I of Kowalewski top in a magnetic field”, J. Math. Sci., 149:1 (2008), 922–946  mathnet  crossref  mathscinet  zmath  elib
    18. D. B. Zot'ev, “Contact degeneracies of closed 2-forms”, Sb. Math., 198:4 (2007), 491–520  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    19. Sagar Chakraborty, J. K. Bhattacharjee, “Effects of nondenumerable fixed points in finite dynamical systems”, Chaos, 18:1 (2008), 013124  crossref  mathscinet  isi  elib
    20. 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
    21. P. E. Ryabov, M. P. Kharlamov, “Analiticheskaya klassifikatsiya osobennostei obobschennogo volchka Kovalevskoi”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 2, 19–28  mathnet  elib
    22. M. P. Kharlamov, “Topologicheskii analiz i bulevy funktsii: I. Metody i prilozheniya k klassicheskim sistemam”, Nelineinaya dinam., 6:4 (2010), 769–805  mathnet
    23. M. P. Kharlamov, “Topologicheskii analiz i bulevy funktsii: II. Prilozheniya k novym algebraicheskim resheniyam”, Nelineinaya dinam., 7:1 (2011), 25–51  mathnet
    24. 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
    25. 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
    26. 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
    27. Xiang Zhang, “Comment on “On the polynomial integrability of the Kirchoff equations, Physica D 241 (2012) 1417–1420””, Physica D: Nonlinear Phenomena, 2013  crossref
    28. 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
    29. S.S.. Zub, “Stable orbital motion of magnetic dipole in the field of permanent magnets”, Physica D: Nonlinear Phenomena, 2014  crossref
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