ar X iv : a st ro - p h / 03 12 55 1 v 2 1 8 M ay 2 00 4 The three - body problem and the Hannay angle

The Hannay angle has been previously studied for a celestial circular restricted three-body system by means of an adiabatic approach. In this work, three main results are obtained. Firstly, a formal connection between perturbation theory and the Hamiltonian adiabatic approach, both leading to the Hannay angle; it is thus emphasised that this effect is already contained in classical celestial mechanics, although not defined nor evaluated separately yet. Secondly, a more general expression of the Hannay angle, valid for an action dependent potential; such generalised expression takes into account that the restricted three-body problem is a time-dependent, two degrees of freedom, problem even when restricted to the circular motion of the test body. As a consequence, (some of) the eccentricity terms cannot be neglected a priori. Thirdly, a numerical estimate for the Earth and a geostationary satellite, adiabatically driven by Jupiter and the Moon, respectively and a conceptual description of a possible measurement. We also point out errors in a previous derivation of the Hannay angle for the circular restricted three-body problem, with an action independent potential.