New Methods of Isochrone Mechanics

TL;DR

This paper revisits the classical problem of isochrone potentials using Hamiltonian mechanics, action-angle coordinates, and Birkhoff normal forms to deepen understanding and establish new proofs of fundamental properties.

Contribution

It introduces a Hamiltonian mechanics approach with action-angle coordinates and Birkhoff normal forms to analyze isochrone potentials, providing a new proof of the isochrone theorem.

Findings

Validates Kepler equation and eccentric anomaly for all isochrone orbits

Provides a self-consistent proof of the isochrone theorem

Shows how celestial mechanics results are encoded in the formalism

Abstract

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.