Advertisement

Three and four-body systems in one dimension: Integrability, superintegrability and discrete symmetries

  • Claudia ChanuEmail author
  • Luca Degiovanni
  • Giovanni Rastelli
Article

Abstract

Families of three-body Hamiltonian systems in one dimension have been recently proved to be maximally superintegrable by interpreting them as one-body systems in the three-dimensional Euclidean space, examples are the Calogero, Wolfes and Tramblay Turbiner Winternitz systems. For some of these systems, we show in a new way how the superintegrability is associated with their dihedral symmetry in the three-dimensional space, the order of the dihedral symmetries being associated with the degree of the polynomial in the momenta first integrals. As a generalization, we introduce the analysis of integrability and superintegrability of four-body systems in one dimension by interpreting them as one-body systems with the symmetries of the Platonic polyhedra in the four-dimensional Euclidean space. The paper is intended as a short review of recent results in the sector, emphasizing the relevance of discrete symmetries for the superintegrability of the systems considered.

Keywords

superintegrability higher-degree first integrals discrete symmetries Tremblay-Turbiner-Winterniz system 

MSC2010 numbers

70H06 70F07 70F10 37J35 37J15 

References

  1. 1.
    Evans, N. W., Superintegrability in Classical Mechanics, Phys. Rev. A, 1990, vol. 41, no. 10, pp. 5666–5676.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chanu, C., Degiovanni, L., and Rastelli, G., Superintegrable Three-Body Systems on the Line, J. Math. Phys., 2008, vol. 49, no. 11, 112901 (10 pp.).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Multiparticle Systems: The Algebra of Integrals and Integrable Cases, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18–41.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benenti, S., Chanu, C., and Rastelli, G., The Super-Separability of the Three-Body Inverse-Square Calogero System, J. Math. Phys., 2000, vol. 41, no. 7, pp. 4654–4678.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chanu, C., Degiovanni, L., and Rastelli, G., Polynomial Constants of Motion for Calogero-Type Systems in Three Dimensions, J. Math. Phys., 2011, vol. 52, 032903 (7 pp.).MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chanu, C., Degiovanni, L., and Rastelli, G., First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator, SIGMA, 2011, vol. 7, Paper 038 (12 pp.).Google Scholar
  7. 7.
    Maciejewski, A. J., Przybylska, M., and Yoshida, H., Necessary Conditions for Classical Super-Integrability of a Certain Family of Potentials in Constant Curvature Spaces, J. Phys. A, 2010, vol. 43, no. 38, 382001 (15 pp.).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kalnins, E. G., Kress, J. M., and Miller, W., Jr., Tools for Verifying Classical and Quantum Superintegrability, SIGMA, 2010, vol. 6, Paper 066 (23 pp.).Google Scholar
  9. 9.
    Tremblay, F., Turbiner, A.V., and Winternitz, P., An Infinite Family of Solvable and Integrable Quantum Systems on a Plane, J. Phys. A, 2009, vol. 42, no. 24, 242001 (10 pp.).MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tremblay, F., Turbiner, A.V., and Winternitz, P., Periodic Orbits for an Infinite Family of Classical Superintegrable Systems, J. Phys. A, 2010, vol. 43, no. 1, 015202 (14 pp.).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kalnins, E. G., Kress, J. M., and Miller, W., Jr., Families of Classical Subgroup Separable Superintegrable Systems, J. Phys. A, 2010, vol. 43, 092001 (8 pp.).MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kalnins, E. G., Miller, W., Jr., and Pogosyan, G. S., Superintegrability and Higher Order Constants for Classical and Quantum Systems, Phys. Atom. Nucl., 2011, vol. 74, pp. 914–918.CrossRefGoogle Scholar
  13. 13.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Superintegrable System on a Sphere with the Integral of Higher Degree, Regul. Chaotic Dyn., 2009, vol. 14, no. 6, pp. 615–620.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Tsiganov, A.V., Leonard Euler: Addition Theorems and Superintegrable Systems, Regul. Chaotic Dyn., 2009, vol. 14, no. 3, pp. 389–406.MathSciNetCrossRefGoogle Scholar
  15. 15.
    Maciejewski, A. J., Przybylska, M., and Tsiganov, A.V., On Algebraic Construction of Certain Integrable and Super-Integrable Systems, Physica D, 2011, vol. 240, pp. 1426–1448.CrossRefGoogle Scholar
  16. 16.
    Chanu, C., Degiovanni, L., and Rastelli, G., Superintegrable Three-Body Systems in One Dimension and Generalizations, arxiv: 0907.5288v1, 2009.Google Scholar
  17. 17.
    Hakobyan, T., Nersessian, A., and Yeghikian, V., The Cuboctahedric Higgs Oscillator from the Rational Calogero Model, J. Phys. A, 2009, vol. 42, no. 20, 205206 (11 pp.).MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hakobyan, T., Krivonos, S., Lechtenfeld, O., and Nersessian, A., Hidden Symmetries of Integrable Conformal Mechanical Systems, Phys. Lett. A, 2010, vol. 374, no. 6, pp. 801–806.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wojciechowski, S., Superintegrability of the Calogero-Moser System, Phys. Lett. A, 1983, vol. 95, pp. 279–281.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rastelli, G., Searching for Integrable Hamiltonian Systems with Platonic Symmetries, arXiv:1001.0752, 2010.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  • Claudia Chanu
    • 1
    Email author
  • Luca Degiovanni
    • 2
  • Giovanni Rastelli
    • 2
  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly
  2. 2.Formerly at Dipartimento di MatematicaUniversità di TorinoTorinoItaly

Personalised recommendations