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On Action-angle coordinates and the Poincaré Coordinates

Féjoz, Jacques (2013), On Action-angle coordinates and the Poincaré Coordinates, Regular and Chaotic Dynamics, 18, 6, p. 708-723. http://dx.doi.org/10.1134/S1560354713060105

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Type
Article accepté pour publication ou publié
Date
2013
Journal name
Regular and Chaotic Dynamics
Volume
18
Number
6
Publisher
Springer
Pages
708-723
Publication identifier
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Author(s)
Féjoz, Jacques
Abstract (EN)
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.
Subjects / Keywords
perturbation theory; integrability; first integral; planetary problem; Poincaré coordinates; two-body problem; Kepler problem; adiabatic invariants; Liouville-Arnold theorem; action-angle coordinates; Lagrangian fibration; Hamiltonian system