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Journal of Mathematical Sciences

, Volume 213, Issue 5, pp 769–785 | Cite as

Degenerately Integrable Systems

  • N. Reshetikhin
Article

The subject of this paper is degenerate integrability in Hamiltonian mechanics. We start with a short survey of degenerate integrability. The first section contains basic notions. It is followed by a number of examples which include the Kepler system, Casimir models, spin Calogero models, spin Ruijsenaars models, and integrable models on symplectic leaves of Poisson Lie groups. The new results are degenerate integrability of relativistic spin Ruijsenaars and Calogero–Moser systems and the duality between them. Bibliography: 30 titles.

Keywords

Modulus Space Conjugacy Class Poisson Structure Poisson Algebra Coadjoint Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Alekseev, E. Meinrenken, and C. Woodward, “Group-valued equivariant localization,” Invent. Math., 140, 327–350 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    V. Ayadi, L. Feher, and T.F. Gorbe, “Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals,” J. Geom. Symmetry Phys., 27, 27–44 (2012).MathSciNetzbMATHGoogle Scholar
  3. 3.
    F. Calogero, “Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials,” J. Math. Phys., 12, 419–436 (1971).CrossRefMathSciNetGoogle Scholar
  4. 4.
    B. Enriquez and V. Rubtsov, “Hitchin systems, higher Gaudin operators, and R-matrices,” Math. Res. Lett., 3, 343–357 (1996).CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    L. Feher and C. Klimyk, “Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction,” Nucl.Phys., B860, 464–515 (2012).Google Scholar
  6. 6.
    L. Feher and C. Klimcik, “Poisson–Lie interpretation of trigonometric Ruijsenaars duality,” Commun. Math. Phys., 301, 55–104 (2011).CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    L. Feher and B. G. Pusztai, “Twisted spin Sutherland models from quantum Hamiltonian reduction,” J. Phys. A, 41, 194009 (2008).CrossRefMathSciNetGoogle Scholar
  8. 8.
    L. Feher and B. G. Pusztai, “Generalized spin Sutherland systems revisited,” Nucl. Phys., B893, 236–256 (2015).CrossRefMathSciNetGoogle Scholar
  9. 9.
    V. Fock, A. Gorsky, N. Nekrasov, and A. Rubtsov, “Dualities in integrable gauge theories,” JHEP, 0007 (2000), 028.CrossRefMathSciNetGoogle Scholar
  10. 10.
    V. Fock, “Zur Theorie Des Wasserstoffatoms,” Z. Physik, 98, 145 (1935).CrossRefGoogle Scholar
  11. 11.
    V. Fock and A. A. Rosly, “Flat connections and polyubles,” Teor. Mat. Fiz., 95, 228–238 (1993).CrossRefMathSciNetGoogle Scholar
  12. 12.
    A. S. Mischenko and A. T. Fomenko, “Generalized Liouville method or integrating Hamiltonian systems,” Functs. Analiz Prilozh., 12, 46–56 1978.Google Scholar
  13. 13.
    J. Frish, V. Mandrosov, and Y. A. Smorodinsky, M. Uhlir, and P. Winternitz. “On higher symmetries in quantum mechanics,” Physics Letters, 16, 354–356 (1965).Google Scholar
  14. 14.
    M. I. Gekhtman and M. Z. Shapiro. “Noncommutative and commutative integrability of generic Toda flow in simple Lie algberas,” Comm. Pure Appl. Math., 52, 53–84 (1999).CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    J. Gibbons and T. Hermsen, “A generalization of the Calogero–Moser system,” Physica, 11D, 337 (1984).MathSciNetGoogle Scholar
  16. 16.
    D. Kazhdan, B. Kostant, and S. Sternberg. “Hamiltonian group actions and dynamical systems of Calogero type,” Comm. Pure Appl. Math., 31, 481–507 (1978).CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    I. Krichever, O. Babelon, E. Billey, and M. Talon, “Spin Generalization of Calogero–Moser system and the matrix KP equation,” Translations of Amer. Math. Soc., Series 2, Vol. 170, Advances in Mathematical Science, Topics in Topology and Math. Phys., hep-th/9411160 1975.Google Scholar
  18. 18.
    L. C. Li and P. Xu, “Spin Calogero–Moser systems associated with simple Lie algebras,” C. R. Acad. Sci. Paris, Serie I, 331, 55–61 (2000).Google Scholar
  19. 19.
    J. Moser, “Three integrable Hamiltonian systems connected with isospectral deformations,” Advances Math., 16, 197–220 (1975).CrossRefzbMATHGoogle Scholar
  20. 20.
    N. N. Nekhoroshev, “Action-angle variables and their generalizations,” Trans. Moscow Math. Soc., 26, 180–197 (1972).Google Scholar
  21. 21.
    N. Nekrasov, “Holomorphyc bundles and many-body systems,” CMP, 180, 587–604 (1996).MathSciNetzbMATHGoogle Scholar
  22. 22.
    M. A. Olshanetsky and A. M. Perelomov, “Quantum integrable systems related to Lie algebras,” Phys. Rept., 94, 313–404 (1983).CrossRefMathSciNetGoogle Scholar
  23. 23.
    W. Pauli, “On the hydrogen spectrum from the standpoint of the new quantum mechanics,” Zeitschrift fur Physik, 36, 336–363 (1926).CrossRefzbMATHGoogle Scholar
  24. 24.
    N. Reshetikhin, “Integrability of characteristic Hamiltonian systems on simple Lie groups with standard Poisson–Lie structure,” Comm. Math. Phys., 242, 1–29 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    N. Reshetikhin, “Degenerate integrability of the spin Calogero–Moser systems and the duality with the spin Ruijsenaars systems,” Lett. Math. Phys., 63, 55–71 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    S. Ruijsenaars, “Systems of Calogero–Moser type,” in: Proc. of the 1994 CRM Banff Summer School on Particles and Fields, Springer (1999), pp. 251–352Google Scholar
  27. 27.
    M. Semenov-Tian-Shansky, “Dressing transformations and Poisson group actions,” Publ. Res. Inst. Math. Sci., 21, 1237–1260 (1985).CrossRefMathSciNetGoogle Scholar
  28. 28.
    B. Sutherland, “Exact results for a many-body problem in one dimension II,” Phys. Rev. A, 5, 1372–1376 (1972).CrossRefGoogle Scholar
  29. 29.
    S. Wojciechowski, “Superintegrability of the Calogero–Moser system,” Phys. Lett. A, 95, 279 (1983).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.University of AmsterdamAmsterdamThe Netherlands
  3. 3.ITMO UniversitySt.PetersburgRussia

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