New Methods of Isochrone Mechanics - ENSTA Paris - École nationale supérieure de techniques avancées Paris Accéder directement au contenu
Article Dans Une Revue Journal of Mathematical Physics Année : 2021

New Methods of Isochrone Mechanics

Paul Ramond
  • Fonction : Auteur
Jérôme Perez

Résumé

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.

Dates et versions

hal-03196975 , version 1 (13-04-2021)

Identifiants

Citer

Paul Ramond, Jérôme Perez. New Methods of Isochrone Mechanics. Journal of Mathematical Physics, 2021, 62, pp.112704. ⟨10.1063/5.0056957⟩. ⟨hal-03196975⟩
88 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More