Geometric Derivation of the Delaunay Variables and Geometric Phases

Authors: Eui Chang, D.¹; Marsden, J.E.²

Source: Celestial Mechanics and Dynamical Astronomy, Volume 86, Number 2, June 2003 , pp. 185-208(24)

Publisher: Springer

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Abstract:

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T³ 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 S¹ 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 J₂-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J₂-Hamiltonian is a collective Hamiltonian of the T³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J₂ effect as geometric phases.

Keywords: Kepler vector field; derivation of variables; orbits dynamics and phases

Document Type: Research article

Affiliations: 1: Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106, U.S.A., e-mail: dchang@engineering.ucsb.edu 2: Control and Dynamical Systems 107-81, California Institute of Technology, Pasadena, CA 91125, U.S.A., e-mail: marsden@cds.caltech.edu

Publication date: 2003-06-01