Abstract:
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 "∮γpdq". 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.
\Bibitem{Fej13}
\by Jacques F\'ejoz
\paper On Action-angle Coordinates and the Poincaré Coordinates
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 6
\pages 703--718
\mathnet{http://mi.mathnet.ru/rcd165}
\crossref{https://doi.org/10.1134/S1560354713060105}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3146588}
\zmath{https://zbmath.org/?q=an:1290.01001}
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This publication is cited in the following 13 articles:
Donato Scarcella, “Weakly asymptotically quasiperiodic solutions for time-dependent Hamiltonians with a view to celestial mechanics”, Journal of Differential Equations, 2025
Andrew Clarke, Jacques Fejoz, Marcel Guardia, “Why are inner planets not inclined?”, Publ.math.IHES, 2024
Ramond P., Perez J., “New Methods of Isochrone Mechanics”, J. Math. Phys., 62:11 (2021), 112704
Jackman C., “Secular Dynamics For Curved Two-Body Problems”, J. Dyn. Differ. Equ., 2021
Barbieri S., Niederman L., “Sharp Nekhoroshev Estimates For the Three-Body Problem Around Periodic Orbits”, J. Differ. Equ., 268:7 (2020), 3749–3780
G. Pinzari, Perihelia reduction and global Kolmogorov tori in the planetary problem, Mem. Am. Math. Soc., 255, no. 1218, 2018, v+92 pp.
Nguyen Tien Zung, “A conceptual approach to the problem of action-angle variables”, Arch. Ration. Mech. Anal., 229:2 (2018), 789–833
J. Laskar, “Andoyer construction for Hill and Delaunay variables”, Celest. Mech. Dyn. Astron., 128:4 (2017), 475–482
A. Boscaggin, R. Ortega, “Periodic solutions of a perturbed Kepler problem in the plane: from existence to stability”, J. Differ. Equ., 261:4 (2016), 2528–2551
A. Kiesenhofer, E. Miranda, G. Scott, “Action-angle variables and a KAM theorem for b-Poisson manifolds”, J. Math. Pures Appl., 105:1 (2016), 66–85
Yu. A. Grigoryev, A. V. Tsiganov, “On bi-Hamiltonian formulation of the perturbed Kepler problem”, J. Phys. A-Math. Theor., 48:17 (2015), 175206
G. Pinzari, “Global Kolmogorov tori in the planetary N-body problem. Announcement of result”, Electron. Res. Announc. Math. Sci., 22 (2015), 55–75
Gabriella Pinzari, “Global Kolmogorov tori in the planetary N-body problem. Announcement of result”, ERA-MS, 22 (2015), 55
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