Communications in Mathematical Physics

, Volume 83, Issue 1, pp 83–106 | Cite as

Kowalewski's asymptotic method, Kac-Moody lie algebras and regularization

  • M. Adler
  • P. van Moerbeke


We use an effective criterion based on the asymptotic analysis of a class of Hamiltonian equations to determine whether they are linearizable on an abelian variety, i.e., solvable by quadrature. The criterion is applied to a system with Hamiltonian
$$H = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\sum\limits_{i = 1}^l {p_i^2 } + \sum\limits_{i = 1}^{l + 1} {\exp \left( {\sum\limits_{j = 1}^l {N_{ij} x_j } } \right)} ,$$
parametrized by a real matrixN=(N ij ) of full rank. It will be solvable by quadrature if and only if for allij, 2(N NT) ij (N N T ) jj −1 is a nonpositive integer, i.e., the interactions correspond to the Toda systems for the Kac-Moody Lie algebras. The criterion is also applied to a system of Gross-Neveu.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math.38, 267–317 (1980)Google Scholar
  2. 2.
    Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. Math.38, 318–379 (1980)Google Scholar
  3. 3.
    Carter, R.W.: Simple groups of Lie type. London, New York: Wiley 1972Google Scholar
  4. 4.
    Helgason, S.: Differential geometry, Lie groups and symmetric spaces. New York: Academic Press 1978Google Scholar
  5. 5.
    Hille, E.: Ordinary differential equations in the complex domain. New York: Wiley-Interscience 1976Google Scholar
  6. 6.
    Humphreys, J.E.: Introduction to Lie algebras and representation theory. Berlin, Heidelberg, New York: Springer 1972Google Scholar
  7. 7.
    Kowalewski, S.: Sur le probleme de la rotation d'un corps solide autour d'un point fixe. Acta Math.12, 177–232 (1889)Google Scholar
  8. 8.
    Kowalewski, S.: Sur une propriété du système d'équations différentielles qui définit la rotation d'un corps solide autour d'un point fixe. Acta Math.14, 81–93 (1891)Google Scholar
  9. 9.
    McKean, H.P.: Theta functions, solitons and singular curves, partial differential equations and geometry, pp. 237–254. New York: M. Dekker, Inc. 1979Google Scholar
  10. 10.
    Bogoyavlensky, O.I.: On perturbations of the periodic Toda lattices. Commun. Math. Phys.51, 201–209 (1976)Google Scholar
  11. 11.
    Shankar, R.: A model that acquires integrability andO(2N) invariance at a critical coupling. Preprint Yale University YTP 81-06Google Scholar
  12. 12.
    Adler, M., van Moerbeke, P.: The algebraic integrability of geodesic flow on SO(4). Invent. Math. (to appear) (1982)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • M. Adler
    • 1
  • P. van Moerbeke
    • 2
    • 3
  1. 1.Brandeis UniversityWalthamUSA
  2. 2.University of LouvainLouvain-la-NeuveBelgium
  3. 3.Brandeis UniversityWalthamUSA

Personalised recommendations