New methods of isochrone mechanics

Abstract

Isochrone potentials are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Whereas all previous results on isochrone mechanics have been established using classical analysis and geometry, in this article, we revisit the isochrone problem of motion using tools from Hamiltonian dynamical systems. In particular, we (1) solve the problem of motion using a well-adapted set of angle-action coordinates and generalize the notion of eccentric anomaly to all isochrone orbits, and (2) we construct the Birkhoff normal form for a particle orbiting a generic radial potential and examine its Birkhoff invariants to prove that the class of isochrone potentials is in correspondence with parabolas in the plane. Along the way, several fundamental results of celestial mechanics, such as the Bertrand theorem or the Kepler equation and laws, are obtained as special cases of more general properties characterizing isochrone mechanics.