Isochrony in 3D radial potentials. From Michel H\'enon ideas to
  isochrone relativity: classification, interpretation and applications
  Revisiting and extending an old idea of Michel H\'enon, we geometrically and
algebraically characterize the whole set of isochrone potentials. Such
potentials are fundamental in potential theory. They appear in spherically
symmetrical systems formed by a large amount of charges (electrical or
gravitational) of the same type considered in mean-field theory. Such
potentials are defined by the fact that the radial period of a test charge in
such potentials, provided that it exists, depends only on its energy and not on
its angular momentum.
  Our characterization of the isochrone set is based on the action of a real
affine subgroup on isochrone potentials related to parabolas in the
$\mathbb{R}^2$ plane. Furthermore, any isochrone orbits are mapped onto
associated Keplerian elliptic ones by a generalization of the Bohlin
transformation. This mapping allows us to understand the isochrony property of
a given potential as relative to the reference frame in which its parabola is
represented. We detail this isochrone relativity in the special relativity
formalism.
  We eventually exploit the completeness of our characterization and the
relativity of isochrony to propose a deeper understanding of general symmetries
such as Kepler's Third Law and Bertrand's theorem.
