Geometric Derivation of the Delaunay Variables and Geometric Phases

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.