Liouville-Arnold Integrability for Scattering Under Cone Potentials
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.
Unable to display preview. Download preview PDF.
- [A]Arnold, V. I. (ed.) (1988). Encyclopaedia of mathematical sciences 3, Dynamical Systems III. Springer Verlag, Berlin.Google Scholar
- [GZ1]Gorni, G., & Zampieri, G. (1989). Complete integrability for Hamiltonian systems with a cone potential. To appear in J. Diff. Equat.Google Scholar
- [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
- [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
- [He]Herbst (1974). Classical scattering with long range forces. Comm. Math. Phys. 35, pp. 193–214.Google Scholar
- [MNO]Moauro, V., Negrini, P., & Oliva, W.M. (1989). Analytic integrability for a class of cone potential mechanical systems. In preparation.Google Scholar
- [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
- [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