The three-body problem and the Hannay angle

Abstract

The Hannay angle has been previously studied for a celestial circular restricted three-body system by means of an adiabatic approach. In the present work, three main results are obtained. First, a formal connection between perturbation theory and the Hamiltonian adiabatic approach shows that both lead to the Hannay angle; it is thus emphasized that this effect is already contained in classical celestial mechanics, although not yet defined nor evaluated separately. Second, a more general expression of the Hannay angle, valid for an action-dependent potential is given; such a generalized expression takes into account the fact that the circular restricted three-body problem (RTBP) is a time-dependent problem with two degrees of freedom even when restricted to the circular motion of the test body. Consequently (some of) the eccentricity terms cannot be neglected a priori. Third, we present a new numerical estimate for the Earth adiabatically driven by Jupiter. We also point out errors in a previous derivation of the Hannay angle for the circular RTBP, with an action-independent potential.