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In classical mechanics, the Kepler potential and the Harmonic potential sharethe following remarkable property: in either of these potentials, a bound testparticle orbits with a radial period that is independent of its angularmomentum. For this reason, the Kepler and Harmonic potentials are called\it{isochrone}. In this paper, we solve the following general problem: arethere any other isochrone potentials, and if so, what kind of orbits do theycontain? To answer these questions, we adopt a geometrical point of viewinitiated by Michel H\'enon in 1959, in order to explore and classifyexhaustively the set of isochrone potentials and isochrone orbits. Inparticular, we provide a geometric generalization of Kepler's third law, andgive a similar law for the apsidal angle, for any isochrone orbit. We alsorelate the set of isochrone orbits to the set of parabolae in the plane underlinear transformations, and use this to derive an analytical parameterizationof any isochrone orbit. Along the way we compare our results to known ones,pinpoint some interesting details of this mathematical physics problem, andargue that our geometrical methods can be exported to more generic orbits inpotential theory.

The Geometry of Isochrone Orbits: from Archimedes' parabolae to Kepler's third law