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Geometric Derivation of the Delaunay Variables and Geometric Phases

Chang, Dong Eui and Marsden, Jerrold E. (2003) Geometric Derivation of the Delaunay Variables and Geometric Phases. Celestial Mechanics and Dynamical Astronomy, 86 (2). pp. 185-208. ISSN 0923-2958.

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We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T^3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S ^1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J _(2-)dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J _(2-)Hamiltonian is a collective Hamiltonian of the T^3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J _2 effect as geometric phases.

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Additional Information:© 2003 Kluwer Academic Publishers. Received: 20 December 2001; revised: 27 September 2002; accepted: 2 November 2002. Dedicated to Klaus Kirchgässner on the occasion of his 70th birthday. This research was partially supported by the Humboldt Foundation and a Max-Planck Research Award and the California Institute of Technology.
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Humboldt FoundationUNSPECIFIED
Max-Planck Research AwardUNSPECIFIED
California Institute of TechnologyUNSPECIFIED
Subject Keywords:Kepler vector field, derivation of variables, orbits dynamics and phases
Record Number:CaltechAUTHORS:20100730-131231526
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19232
Deposited By: Ruth Sustaita
Deposited On:02 Aug 2010 21:43
Last Modified:26 Dec 2012 12:16

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