Geometric Derivation of the Delaunay Variables and Geometric Phases

Chang, D. E. and J. E. Marsden

Dedicated to Klaus Kirchgässner on the ocassion of his 70th birthday

Abstract:

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus $\mathbb {T}$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 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 $\mathbb {T}$³ momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J₂ effect as geometric phases.