Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 046, 23 pages      math-ph/0703045
Contribution to the Vadim Kuznetsov Memorial Issue

Qualitative Analysis of the Classical and Quantum Manakov Top

Evguenii Sinitsyn a and Boris Zhilinskii b
a) Physics Department, Tomsk State University, 634050 Tomsk, Russia
b) Université du Littoral, UMR du CNRS 8101, 59140 Dunkerque, France

Received 20 October, 2006, in final form 19 January, 2007; Published online March 13, 2007

Qualitative features of the Manakov top are discussed for the classical and quantum versions of the problem. Energy-momentum diagram for this integrable classical problem and quantum joint spectrum of two commuting observables for associated quantum problem are analyzed. It is demonstrated that the evolution of the specially chosen quantum cell through the joint quantum spectrum can be defined for paths which cross singular strata. The corresponding quantum monodromy transformation is introduced.

Key words: Manakov top; energy-momentum diagram; monodromy.

pdf (781 kb)   ps (774 kb)   tex (1004 kb)


  1. Adler M., van Moerbeke P., The Kowalewski and Hénon-Heiles motions as Manakov geodesic flows on SO(4) - a two-dimensional family of Lax pairs, Comm. Math. Phys. 113 (1988), 659-700.
  2. Audin M., Spinning tops, Cambridge University Press, Cambridge, 1996, Chapter 4.
  3. Bolsinov A.V., Fomenko A.T., Integrable Hamiltonian systems. Geometry, topology, classification, Chapman & Hall/CRC London, 2004, Section 14.
  4. Cejnar P., Macek M., Heinze S., Jolie J., Dobes J., Monodromy and excited-state quantum phase transitions in integrable systems: collective vibrations of nuclei, J. Phys. A: Math. Gen. 39 (2006), L515-L521.
  5. Child M.S., Quantum monodromy and molecular spectroscopy, Adv. Chem. Phys., to appear.
  6. Colin de Verdière Y., Vu Ngoc S., Singular Bohr-Sommerfeld rules for 2D integrable systems, Ann. Sci. Ècole Norm. Sup. (4) 36 (2003), 1-55, math.AP/0005264.
  7. Cushman R.H., Bates L.M., Global aspects of classical integrable systems, Birkhäuser, Basel, 1997.
  8. Cushman R.H., Sadovskii D., Monodromy in the hydrogen atom in crossed fields, Phys. D 142 (2000), 166-196.
  9. Davison C.M., Dullin H.R., Bolsinov A.V., Geodesics on the ellipsoid and monodromy, math-ph/0609073.
  10. Davison C.M., Dullin H.R., Geodesic flow on three dimensional ellipsoids with equal semi-axes, math-ph/0611060.
  11. Duistermaat J.J., On global action angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.
  12. Efstathiou K., Cushman R.H., Sadovskii D.A., Fractional monodromy in the 1:-2 resonance, Adv. Math. 209 (2007), 241-273.
  13. Efstathiou K., Sadovskii D., Zhilinskii B., Analysis of rotation-vibration relative equilibria on the example of a tetrahedral four atom molecule, SIAM J. Appl. Dyn. Syst. 3 (2004), 261-351.
  14. Grondin L., Sadovskii D., Zhilinskii B., Monodromy in systems with coupled angular momenta and rearrangement of bands in quantum spectra, Phys. Rev. A 65 (2002), 012105, 15 pages.
  15. Kalnins E. G., Miller W.Jr., Winternitz P., The group O(4), separation of variables and the hydrogen atom, SIAM J. Appl. Math. 30 (1976), 630-664.
  16. Komarov I.V., Kuznetsov V.B., Quantum Euler-Manakov top on the 3-sphere S3, J. Phys. A: Math. Gen. 24 (1991), L737-L742.
  17. Leung N.C., Symington M., Almost toric symplectic four-manifolds, math.SG/0312165.
  18. Manakov S.V., Note on the integration of Euler's equation of the dynamics of an N dimensional rigid body, Funct. Anal. Appl. 11 (1976), 328-329.
  19. Michel L., Points critique des fonctions invariantes sur une G-varieté, C. R. Math. Acad. Sci. Paris 272 (1971), 433-436.
  20. Michel L., Zhilinskii B.I., Symmetry, invariants, and topology. I. Basic tools, Phys. Rep. 341 (2001), 11-84.
  21. Nekhoroshev N.N., Action-angle variables and their generalizations, Tr. Mosk. Mat. Obs. 26 (1972), 180-198.
  22. Nekhoroshev N. N., Sadovskií D.A., Zhilinskií B.I., Fractional monodromy of resonant classical and quantum oscillators, C. R. Math. Acad. Sci. Paris 335 (2002), 985-988.
  23. Nekhoroshev N. N., Sadovskii D., Zhilinskii B., Fractional Hamiltonian monodromy, Ann. Henri Poincaré 7 (2006), 1099-1211.
  24. Oshemkov A.A., Topology of isoenergy surfaces and bifurcation diagrams for integrable cases of rigid body dynamics on so(4), Uspekhi Mat. Nauk 42 (1987), 199-200.
  25. Perelomov A.M., Motion of four-dimensional rigid body around a fixed point: an elementary approach. I, math-ph/0502053.
  26. Sadovskii D., Zhilinskii B., Group theoretical and topological analysis of localized vibration-rotation states, Phys. Rev. A 47 (1993), 2653-2671.
  27. Sadovskii D., Zhilinskii B., Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A 256 (1999), 235-244.
  28. Sadovskii D., Zhilinskii B., Quantum monodromy, its generalizations and molecular manifestations, Mol. Phys. 104 (2006), 2595-2615.
  29. Sadovskii D., Zhilinskii B., Hamiltonian systems with detuned 1:1:2 resonance. Manifestations of bidromy, Ann. Physics 322 (2007), 164-200.
  30. Symington M., Four dimensions from two in symplectic topology, in Topology and Geometry of Manifolds (2001, Athens, GA), Proc. Symp. Pure Math., Vol. 71, AMS, Providence, RI, 2003, 153-208, math.SG/0210033.
  31. Winnewisser M., Winnewisser B., Medvedev I., de Lucia F.C., Ross S.C., Bates L.M., The hidden kernel of molecular quasi-linearity: quantum monodromy, J. Mol. Structure 798 (2006), 1-26.
  32. Vu Ngoc S., Quantum monodromy in integrable systems, Comm. Math. Phys. 203 (1999), 465-479.
  33. Vu Ngoc S., Moment polytopes for symplectic manifolds with monodromy, Adv. Math. 208 (2007), 909-934, math.SG/0504165.
  34. Zhilinskii B.I., Symmetry, invariants, and topology. II Symmetry, invariants, and topology in molecular models, Phys. Rep. 341 (2001), 85-171.
  35. Zhilinskii B., Interpretation of quantum Hamiltonian monodromy in terms of lattice defects, Acta Appl. Math. 87 (2005), 281-307.
  36. Zhilinskii B., Hamiltonian monodromy as lattice defect, in Topology in Condensed Matter, Editor M.I. Monastyrsky, Springer, Berlin, 2006, 165-186, quant-ph/0303181.

Previous article   Next article   Contents of Volume 3 (2007)