Liouville-Arnold Integrability for Scattering Under Cone Potentials

  • G. Gorni
  • G. Zampieri
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)


The problem of scattering of particles on the line with repulsive interactions, gives rise to some well-known integrable Hamiltonian systems, for example, the nonperiodic Toda lattice or Calogero’s system. The aim of this note is to outline our researches which proved the integrability of a much larger class of systems, including some that had never been considered, such as the scattering with very-long-range interaction potential. The integrability of all these systems survives any small enough perturbation of the potential in an arbitrary compact set. Our framework is based on the concept of cone potentials, as defined below, which include the scattering on the line as a particular case.


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  1. [A]
    Arnold, V. I. (ed.) (1988). Encyclopaedia of mathematical sciences 3, Dynamical Systems III. Springer Verlag, Berlin.Google Scholar
  2. [Ga]
    Galperin, G.A. (1982) Asymptotic behaviour of particle motion under repulsive forces. Comm. Math. Phys. 84, pp. 547–556.MathSciNetADSCrossRefGoogle Scholar
  3. [Gu1]
    Gutkin, E. (1985). Integrable Hamiltonian with exponential potentials. Physica D 16, pp. 398–404, North Holland, Amsterdam.MathSciNetADSMATHCrossRefGoogle Scholar
  4. [Gu2]
    Gutkin, E. (1988). Regularity of scattering trajectories in Classical Mechanics. Comm. Math. Phys. 119, pp. 1–12.MathSciNetADSMATHCrossRefGoogle Scholar
  5. [GZ1]
    Gorni, G., & Zampieri, G. (1989). Complete integrability for Hamiltonian systems with a cone potential. To appear in J. Diff. Equat.Google Scholar
  6. [GZ2]
    Gorni, G., & Zampieri, G. (1989). Reducing scattering problems under cone potentials to normal form by global canonical transformations. To appear in J. Diff. Equat.Google Scholar
  7. [GZ3]
    Gorni, G., & Zampieri, G. (1989). A class of integrable Hamiltonian systems including scattering of particles on the line with repulsive interactions. Preprint, University of Udine, UDMI/21/89/RR. Submitted.Google Scholar
  8. [He]
    Herbst (1974). Classical scattering with long range forces. Comm. Math. Phys. 35, pp. 193–214.Google Scholar
  9. [Hu]
    Hubacher A. (1989). Classical scattering theory in one dimension. Comm. Math. Phys. 123, pp. 353–375.MathSciNetADSMATHCrossRefGoogle Scholar
  10. [MNO]
    Moauro, V., Negrini, P., & Oliva, W.M. (1989). Analytic integrability for a class of cone potential mechanical systems. In preparation.Google Scholar
  11. [M]
    Moser, J. (1983). Various aspects of integrable Hamiltonian systems. In Dynamical Systems (C.I.M.E. Lectures, Bressanone 1978), pp. 233–290, sec. print., Birkhäuser, Boston.Google Scholar
  12. [OC]
    Oliva, W.M., & Castilla M.S.A.C. (1988). On a class of C -integrable Hamiltonian systems. To appear in Proc. Royal Society Edinburgh.Google Scholar
  13. [S]
    Simon B. (1971). Wave operators for classical particle scattering. Comm. Math. Phys. 23, pp. 37–48.MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • G. Gorni
    • 1
  • G. Zampieri
    • 2
  1. 1.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItaly
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversità di PadovaPadovaItaly

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