On Action-angle Coordinates and the Poincaré Coordinates

    2013, Volume 18, Number 6, pp.  703-718

    Author(s): Féjoz J.

    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 "$\oint_\gamma pdq$". 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.
    Keywords: Hamiltonian system, Lagrangian fibration, action-angle coordinates, Liouville–Arnold theorem, adiabatic invariants, Kepler problem, two-body problem, Poincaré coordinates, planetary problem, first integral, integrability, perturbation theory
    Citation: Féjoz J., On Action-angle Coordinates and the Poincaré Coordinates, Regular and Chaotic Dynamics, 2013, Volume 18, Number 6, pp. 703-718

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