New methods of isochrone mechanics

Isochrone potentials, as defined by Michel H\'enon in the fifties, are spherically symmetric potentials within which a particle orbits with radial period that is independent of its angular momentum. encompass Kepler and harmonic potential, along many other. In this article, we revisit classical problem motion isochrone from point view Hamiltonian mechanics. First, use particularly well-suited set action-angle coordinates to solve dynamics, showing well-known equation eccentric anomaly parametrisation valid for any orbit (and not just Keplerian ellipses). Second, using powerful machinery Birkhoff normal forms, provide self-consistent proof theorem, relates parabolae plane, basis all literature on subject. Along way, show how some fundamental results celestial mechanics such Bertrand theorem Kepler's third law naturally encoded formalism.