Advertisement

Communications in Mathematical Physics

, Volume 83, Issue 3, pp 355–386 | Cite as

Classical and quantum-mechanical systems of Toda lattice type. I

  • Roe Goodman
  • Nolan R. Wallach
Article

Abstract

The structure of the commutant of Laplace operators in the enveloping and “Poisson algebra” of certain generalized “ax +b” groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Hamiltonian System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adler, M.: Invent. Math.50, 219–248 (1979)Google Scholar
  2. 2.
    Adler, M., van Moerbeke, P.: Adv. Math.38, 267–317 (1980)Google Scholar
  3. 3.
    Arnol'd, V.: Ann. Inst. Fourier (Grenoble)16, 319–361 (1966)Google Scholar
  4. 4.
    Arnol'd, V.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  5. 5.
    Bogoyavlensky, O.I.: Commun. Math. Phys.51, 201–209 (1976)Google Scholar
  6. 6.
    Bourbaki, N.: Groupes et algèbres de Lie, Chaps. IV–VI (Éléments de mathématique, Fasc. XXXIV). Paris: Hermann 1968Google Scholar
  7. 7.
    Flaschka, H.: Phys. Rev. B9, 1924–1925 (1974)Google Scholar
  8. 8.
    Goodman, R., Wallach, N.R.: J. Funct. Anal.39, 199–279 (1980)Google Scholar
  9. 9.
    Helgason, S.: Differential geometry and symmetric spaces. New York: Academic Press 1962Google Scholar
  10. 10.
    Hénon, M.: Phys. Rev. B9, 1921–1923 (1974)Google Scholar
  11. 11.
    Kostant, B.: Invent. Math.48, 101–184 (1978)Google Scholar
  12. 12.
    Kostant, B.: Adv. Math.34, 195–338 (1979)Google Scholar
  13. 13.
    Moser, J.: Various aspects of integrable Hamiltonian systems. In: Dynamical systems. Progress in Mathematics, Vol. 8. Boston: Birkhäuser 1980Google Scholar
  14. 14.
    Miščenko, A.S., Fomenko, A.T.: Izv. Akad. Nauk. SSSR Ser. Mat.42, 396–415 (1978); Math. USSR Izv.12, 371–389 (1978)Google Scholar
  15. 15.
    Nelson, E., Stinespring, W.F.: Am. J. Math.81, 547–560 (1959)Google Scholar
  16. 16.
    Olshanetsky, M.A., Perelomov, A.M.: Invent. Math.54, 261–269 (1979)Google Scholar
  17. 17.
    Ratiu, T.: Involution theorems. In: Geometric methods in mathematical physics. Lecture Notes in Mathematics, Vol. 775. Berlin, Heidelberg, New York: Springer 1980Google Scholar
  18. 18.
    Vergne, M.: Bull. Soc. Math. Fr.100, 301–335 (1972)Google Scholar
  19. 19.
    van Moerbeke, P.: Invent. Math.37, 45–81 (1976)Google Scholar
  20. 20.
    Wallach, N.R.: Symplectic geometry and fourier analysis. Lie Groups: History, Frontiers, and Applications. V. Brookline, Massachusetts: Math. Sci. Press 1977Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Roe Goodman
    • 1
  • Nolan R. Wallach
    • 1
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA

Personalised recommendations