Celestial Mechanics and Dynamical Astronomy

, Volume 86, Issue 2, pp 185–208

Geometric Derivation of the Delaunay Variables and Geometric Phases


  • Dong Eui Chang
    • Mechanical and Environmental EngineeringUniversity of California
  • Jerrold E. Marsden
    • Control and Dynamical Systems 107-81California Institute of Technology

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 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 T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J 2 effect as geometric phases.

Kepler vector field derivation of variables orbits dynamics and phases

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© Kluwer Academic Publishers 2003