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