Symmetry in Nonlinear Mathematical Physics - 2009

Francisco Jose Herranz (Universidad de Burgos, Spain)

Symmetries and superintegrability of new N-dimensional nonlinear oscillators from Euclidean systems

We start with the classical harmonic oscillator and Kepler-Coulomb systems on the N-dimensional Euclidean space. By applying the Stackel transform (or coupling-constant metamorphosis) we obtain two families of maximally superintegrable N-dimensional systems; each of them is parametrized by two free real parameters. Furthermore, in spite of the harmonic oscillator or Kepler origin, all the resulting potentials are interpreted as ''intrinsic'' oscillators on spherical symmetric spaces of nonconstant curvature (which are of a nonlinear nature). The resulting set of (2N–2) functionally independent constants of the motion are also explicitly presented. As a byproduct, the Taub-NUT system is recovered as a particular case of the family of nonlinear oscillators coming from the Euclidean Kepler–Coulomb system. Finally, the nonlinear potentials are briefly analysed with respect to the harmonic oscillator one.