Celestial Mechanics and Dynamical Astronomy

, Volume 86, Issue 2, pp 185–208

Geometric Derivation of the Delaunay Variables and Geometric Phases

  • Dong Eui Chang
  • Jerrold E. Marsden

DOI: 10.1023/A:1024174702036

Cite this article as:
Eui Chang, D. & Marsden, J.E. Celestial Mechanics and Dynamical Astronomy (2003) 86: 185. doi:10.1023/A:1024174702036


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.

Kepler vector field derivation of variables orbits dynamics and phases 

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Dong Eui Chang
    • 1
  • Jerrold E. Marsden
    • 2
  1. 1.Mechanical and Environmental EngineeringUniversity of CaliforniaSanta BarbaraU.S.A., e-mail
  2. 2.Control and Dynamical Systems 107-81California Institute of TechnologyPasadenaU.S.A., e-mail

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