On Action-angle coordinates and the Poincaré Coordinates

Jacques Fejoz (1, 2)

1 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants ''$\oint_\gamma p\, dq$''. The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did.

Mots clés: perturbation theory; Hamiltonian system; Lagrangian fibration; action-angle coordinates; Liouville-Arnold theorem; adiabatic invariants; Kepler problem; two-body problem; Poincaré coordinates; planetary problem; first integral; integrability

Domaines: Systèmes dynamiques [math.DS]; Géométrie symplectique [math.SG]

Jacques Fejoz. On Action-angle coordinates and the Poincaré Coordinates. Regular and Chaotic Dynamics, 2013, 18 (6), pp.708-723. ⟨10.1134/S1560354713060105⟩. ⟨hal-00915268⟩