Regular and Chaotic Dynamics

, Volume 21, Issue 5, pp 581–592 | Cite as

The integrable case of Adler–van Moerbeke. Discriminant set and bifurcation diagram

  • Pavel E. Ryabov
  • Andrej A. Oshemkov
  • Sergei V. Sokolov


The Adler–van Moerbeke integrable case of the Euler equations on the Lie algebra so(4) is investigated. For the LA pair found by Reyman and Semenov-Tian-Shansky for this system, we explicitly present a spectral curve and construct the corresponding discriminant set. The singularities of the Adler–van Moerbeke integrable case and its bifurcation diagram are discussed. We explicitly describe singular points of rank 0, determine their types, and show that the momentum mapping takes them to self-intersection points of the real part of the discriminant set. In particular, the described structure of singularities of the Adler–van Moerbeke integrable case shows that it is topologically different from the other known integrable cases on so(4).


integrable Hamiltonian systems spectral curve bifurcation diagram 

MSC2010 numbers

70E05 70E17 37J35 34A05 


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  1. 1.
    Adler, M. and van Moerbeke, P., A New Geodesic Flow on SO(4), in Probability, Statistical Mechanics, and Number Theory, Adv. Math. Suppl. Stud., vol. 9, Orlando, Fla.: Acad. Press, 1986, pp. 81–96.Google Scholar
  2. 2.
    Stekloff, V.A., Sur le movement d’un corps solide ayant une cavité de forme ellipsoidale remplie par un liquide incompressible et sur les variations des latitudes, Ann. Fac. Sci. Toulouse Math. (3), 1909, vol. 1, 145–256.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Borisov, A.V. and Tsygvintsev, A.V., Kovalevskaya’s Method in Rigid Body Dynamics, Appl. Math. Mech., 1997, vol. 61, no. 1, pp. 27–32; see also: Prikl. Mat. Mekh., 1997, vol. 61, no. 1, pp. 30–36.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Borisov, A.V. and Mamaev, I.S., Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics, Regul. Chaotic Dyn., 2016, vol. 21, no. 2, pp. 232–248.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Poincaré, H., Sur la précession des corps déformables, Bull. Astron., 1910, vol. 27, 321–356.Google Scholar
  6. 6.
    Borisov, A.V., Mamaev, I.S., and Sokolov, V.V., A New Integrable Case on so(4), Dokl. Phys., 2001, vol. 46, no. 12, pp. 888–889; see also: Dokl. Ross. Akad. Nauk, 2001, vol. 381, no. 5, pp. 614–615.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sokolov, V.V., One Class of Quadratic so(4) Hamiltonians, Dokl. Math., 2004, vol. 69, no. 1, pp. 108–111; see also: Dokl. Ross. Akad. Nauk, 2004, vol. 394, no. 5, pp. 602–605.MATHGoogle Scholar
  8. 8.
    Tsiganov, A.V., On Integrable Deformation of the Poincaré System, Regul. Chaotic Dyn., 2002, vol. 7, no. 3, pp. 331–336.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Tsiganov, A.V. and Goremykin, O.V., Integrable Systems on so(4) Related to XXX Spin Chains with Boundaries, J. Phys. A, 2004, vol. 37, no. 17, pp. 4843–4849.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Akbarzadeh, R., Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra so(4), Regul. Chaotic Dyn., 2016, vol. 21, no. 1, pp. 1–17.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Akbarzadeh, R. and Haghighatdoost, Gh., The Topology of Liouville Foliation for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra so(4), Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 317–344.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Haghighatdoost, Gh. and Oshemkov, A.A., The Topology of Liouville Foliation for the Sokolov Integrable Case on the Lie Algebra so(4), Sb. Math., 2009, vol. 200, no. 6, pp. 899–921; see also: Mat. Sb., 2009, vol. 200, no. 6, pp. 119–142.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bolsinov, A.V., Borisov, A.V., and Mamaev, I.S., Topology and Stability of Integrable Systems, Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318; see also: Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Borisov, A.V. and Mamaev, I.S., Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 356–371.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Reyman, A.G. and Semenov-Tian-Shansky, M.A., A New Integrable Case of the Motion of the 4-Dimensional Rigid Body, Comm. Math. Phys., 1986, vol. 105, no. 3, pp. 461–472.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mishchenko, A.S. and Fomenko, A.T., Euler Equations on Finite-Dimensional Lie Groups, Izv. Akad. Nauk SSSR Ser. Mat., 1978, vol. 42, no. 2, pp. 396–415, 471 (Russian).MathSciNetMATHGoogle Scholar
  17. 17.
    Mishchenko, A.S. and Fomenko, A.T., Integrability of Euler’s Equations on Semisimple Lie Algebras, Trudy Sem. Vektor. Tenzor. Anal., 1979, vol. 19, 3–94 (Russian).MathSciNetMATHGoogle Scholar
  18. 18.
    Bolsinov, A.V. and Borisov, A.V., Compatible Poisson Brackets on Lie Algebras, Math. Notes, 2002, vol. 72, nos. 1–2, pp. 10–30; see also: Mat. Zametki, 2002, vol. 72, no. 1, pp. 11–34.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bolsinov, A.V. and Oshemkov, A.A., Bi-Hamiltonian Structures and Singularities of Integrable Systems, Regul. Chaotic Dyn., 2009, vol. 14, nos. 4–5, pp. 431–454.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Brailov, Yu.A., Geometry of Translations of Invariants on Semisimple Lie Algebras, Sb. Math., 2003, vol. 194, nos. 11–12, pp. 1585–1598; see also: Mat. Sb., 2003, vol. 194, no. 11, pp. 3–16.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Konyaev, A.Yu., The Bifurcation Diagram and Discriminant of a Spectral Curve of Integrable Systems on Lie Algebras, Sb. Math., 2010, vol. 201, nos. 9–10, pp. 1273–1305; see also: Mat. Sb., 2010, vol. 201, no. 9, pp. 27–60.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Bolsinov, A. and Izosimov, A., Singularities of Bi-Hamiltonian Systems, Comm. Math. Phys., 2014, vol. 331, no. 2, pp. 507–543.MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Izosimov, A., Singularities of Integrable Systems and Algebraic Curves, Int. Math. Res. Notices, 2016, vol. 2016, 50 pp.Google Scholar
  24. 24.
    Lamb, H., Hydrodynamics, 6th ed., New York: Dover, 1945.MATHGoogle Scholar
  25. 25.
    Borisov, A.V. and Mamaev, I.S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
  26. 26.
    Malkin, I.G., Theory of Stability of Motion, Ann Arbor, Mich.: Univ. of Michigan Library, 1958.MATHGoogle Scholar
  27. 27.
    Lerman, L.M. and Umanskiĭ, Ya.L., Structure of the Poisson Action of R 2 on a Four-Dimensional Symplectic Manifold: 1, Selecta Math. Sov., 1987, vol. 6, no. 4, pp. 365–396.MathSciNetGoogle Scholar
  28. 28.
    Bolsinov, A.V. and Fomenko, A.T., Integrable Hamiltonian Systems: Geometry, Topology, Classification, Boca Raton, Fla.: Chapman & Hall, 2004.MATHGoogle Scholar
  29. 29.
    Khorshidi, Kh., The Topology of an Integrable Hamiltonian System for the Steklov Case on the Lie Algebra so(4), Moscow Univ. Math. Bull., 2006, vol. 61, no. 5, pp. 40–44; see also: Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., 2006, no. 5, pp. 58–61.MathSciNetMATHGoogle Scholar
  30. 30.
    Oshemkov, A.A., The Topology of Surfaces of Constant Energy and Bifurcation Diagrams for Integrable Cases of the Dynamics of a Rigid Body on SO(4), Russian Math. Surveys, 1987, vol. 42, no. 6, pp. 241–242; see also: Uspekhi Mat. Nauk, 1987, vol. 42, no. 6(258), pp. 199–200.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Pavel E. Ryabov
    • 1
    • 2
    • 3
  • Andrej A. Oshemkov
    • 4
  • Sergei V. Sokolov
    • 2
  1. 1.Financial UniversityMoscowRussia
  2. 2.Institute of Machines Science, Russian Academy of SciencesMoscowRussia
  3. 3.Moscow Institute of Physics and Technology (State University)Dolgoprudny, Moscow RegionRussia
  4. 4.Lomonosov Moscow State UniversityLeninskie Gory, MoscowRussia

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