Various Aspects of Integrable Hamiltonian Systems

  • J. Moser
Part of the Progress in Mathematics book series (PM, volume 8)


In these informal lecture notes we discuss a number of integrable Hamiltonian systems which have surfaced recently in very different connections. It is our goal to discuss various aspects underlying the integrabil-ity of a system like that of group representation, isospectral deformation and geometrical considerations. Since this subject is still far from being understood or being systematic we discuss a number of examples which are seemingly disconnected. In fact, there are some rather unexpected connections like between the inverse square potential of Calogero (Section 4) and the Korteweg de Vries equation. Here we show a surprising new connection between the geodesics on an ellipsoid and Hill’s equation with finite gap potential.


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  1. [1]
    V. I. Arnold and A. Avez, Problèmes Ergodiquies de la Mécanique Classique, Gauthier-Viliars, Paris, 1967.Google Scholar
  2. [2]
    V. I. Arnold, Mathematical Methods in Classical Mechanics, Moscow, 1974 (Russian). English translation to appear.Google Scholar
  3. [3]
    C. L. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, 1971.CrossRefGoogle Scholar
  4. [4]
    J. Moser, Stable and random motions in dynamical systems, Ann. Math. Studies, 77, 1973.Google Scholar
  5. [1]
    M. Toda, Wave propagation in anharmonic lattices, Jour. Phys. Soc. Japan 23 (1967) 501–506.CrossRefGoogle Scholar
  6. [2]
    M. Henon, Phys. Rev. B9, 1974, 1921–1923CrossRefGoogle Scholar
  7. [3]
    H. Flaschka, The Toda lattice I, Phys. Rev. B9, (1974) 1924–1925.CrossRefGoogle Scholar
  8. [4]
    J. Moser, Finitely many mass points on the line under the influence of an exponential potential — An integrable system, Lecture Notes in Physics 38, Springer, 1975, 467–497.CrossRefGoogle Scholar
  9. [5]
    J. Moser, Three integrable Hamiltonian systems connected with isospectral deformation, Adv. Math. 16 (1975) 197–220.CrossRefGoogle Scholar
  10. [6]
    H. Airault, H. p. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many body problem, Comm. Pure Appl. Math. 30 (1977) 95–148.CrossRefGoogle Scholar
  11. [7]
    M. Adler and J. Moser, On a class of polynomials connected with the Korteweg de Vries equation, Comm. Math. Phys. (1978) 1–30.Google Scholar
  12. [8]
    F. Calogero, Motion of poles and zeroes of special solutions of non-linear and linear differential equations and related “solvable” many-body problems, preprint, Univ. di Roma, 1977.Google Scholar
  13. [9]
    D. V. Choodnovsky, and G. V. Chodnovsky, Pole expansions of nonlinear partial differential equations, Il Nuovo Cimento, 40 B, 2, 1977.Google Scholar
  14. [10]
    M. A. Olshanetsky and A. M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Inv. Math. 37 (1976) 93–109.CrossRefGoogle Scholar
  15. [1]
    V. I. Arnold, Mathematical Methods in Classical Mechanics, Moscow, 1974, (to appear in English translation), in particular Appendix 5.Google Scholar
  16. [2]
    J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetries. Reports on Math. Physics 5, 1974, 121–130.CrossRefGoogle Scholar
  17. [3]
    J. Marsden, Applications to Global Analysis in Mathematical Physics, Publish or Perish, Inc, 1974, in particular Chap. 6.Google Scholar
  18. [4]
    J. M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970.Google Scholar
  19. [5]
    A. A. Kirillov, Elements of the Theory of Representations, Springer, 1976.CrossRefGoogle Scholar
  20. [6]
    M. Adler, On a trace functional for formal pseudodifferential operators and symplectic structure of the Korteweg-de Vries equation, preprint, Univ. Wisc., 1978.Google Scholar
  21. [1]
    D. Kazhdan, B. Kostant and S. Sternberg, Hamiltonian group actions and dynamical systems of Calogero type, to appear, Comm. Pure Appl. Math 1978.Google Scholar
  22. [2]
    M. Adler, Completely Integrable Systems and Symplectic Actions, MRC Report Report # 1830, Univ. Wisconsin, 1978.Google Scholar
  23. [3]
    M. Adler, On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries equation, preprint, to be published.Google Scholar
  24. [4]
    F. Calogero, Solutions of the one dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, Jour. Math. Phys. 12, 1971, 419–436.CrossRefGoogle Scholar
  25. [5]
    J. Moser, Three integrable Hamiltonian systems connected with isoisospectral deformations, Adv. Math. 16, 1975, 197–220.CrossRefGoogle Scholar
  26. [1]
    C. Jacobi, Vorlesungen über Dynamik, Gesammelte Werke, Supplement band, Berlin, 1884.Google Scholar
  27. [2]
    H. Schüth, Stabilität von periodischen Geodätischen auf n-dimensionalen Ellipsoiden, Dissertation, Bonn, 1972.Google Scholar
  28. [3]
    A. Thimm, Integrabilität bien geodätischen Fluss, Diplomarbeit, Bonn, 1976. (In this paper (in Theorem 4.1) it is shown that the geodesic flow on the ellipsoid admits “global” integrals in involution. This fact is evident from our representation of the integrals.)Google Scholar
  29. [4]
    K. Weierstrass, Math. Werke I, pp. 257–266.Google Scholar
  30. [5]
    D. Hilbert and Cohn Vossen, Auschauliche Geometrie, Dover, 1955, p. 197.Google Scholar
  31. [1]
    C. Neumann, De prdblemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur, Journ. reine Angew. Math. 56, 1859, pp. 46–63.CrossRefGoogle Scholar
  32. [2]
    K. Uhlenbeck, Minimal 2-spheres and tori in S (informal preprint, received 1975).Google Scholar
  33. [3]
    R. Devaney, Transversal homoclinic orbits in an integrable system preprint On the separation of Hamilton-Jacobi equations:Google Scholar
  34. [4]
    E. Rosochatius, Über Bewegungen eines Punktes Dissertation at Univ. Göttingen, Druck von Gebr. Unger, Berlin, 1877 (available at Library of the Math. Institut, Göttingen)Google Scholar
  35. [5]
    P. Stäckel, Über die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variablen. Habilitationsschrift, Halle 1891 (available at Library of the Math. Institut, Göttingen).Google Scholar
  36. [1]
    A. A. Dubrovin, V. B. Matveev and S. P. Novikov, Nonlinear equations of Korteweg de Vries type, finite zone linear operators and Abelian varieties, Russ. Math. Surveys 31 (1976) 59–146.CrossRefGoogle Scholar
  37. [2]
    V. A. Marčenko and I. V. Ostrovskii, A characterization of the spectrum of Hill’s operator. Mat. Sbornik 97 (139) 1975, pp. 493–554.CrossRefGoogle Scholar
  38. [3]
    H. Hochstadt, On the determination of Hill’s equation from its spectrum, Arch. Rat. Mech. Anal., Vol. 19 (1965) 353–362.CrossRefGoogle Scholar
  39. [4]
    H.P. McKean and P. van Moerbeke, The spectrum of Hill’s equation, Inventiones Math. 30 (1975) 217–274.CrossRefGoogle Scholar
  40. [5]
    H. P. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976) 14–226.Google Scholar
  41. [6]
    E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977) 321–337.CrossRefGoogle Scholar
  42. [7]
    N. Levinson, The inverse Sturm-Liouville problem, Mat. Tidsskr. B. (1949) 25–30.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1980

Authors and Affiliations

  • J. Moser
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityUSA

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