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. Firstly, a formal connection between perturbation theory and the Hamiltonian adiabatic approach shows that both lead to the Hannay angle; it is thus emphasised that this effect is already contained in classical celestial mechanics, although not yet defined nor evaluated separately. Secondly, a more general expression of the Hannay angle, valid for an action-dependent potential is given; such a 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. Consequently, (some of) the eccentricity terms cannot be neglected {\it a priori}. Thirdly, 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 restricted three-body problem, with an action-independent potential.

11 pages. Accepted by Nonlinearity

Cited by in corpus (8)

• Gravimagnetism, Causality, and Aberration of Gravity in the Gravitational Light-Ray Deflection Experiments
• Intrinsic Dynamical Fluctuation Assisted Symmetry Breaking in Adiabatic Following
• Hierarchical Theory of Quantum Adiabatic Evolution
• Dynamical fluctuations in classical adiabatic processes: General description and their implications
• A semi-relativistic approach to the circular restricted three body problem and a numerical experiment around the 3:2 resonance
• Particle on a Torus Knot: Anholonomy and Hannay Angle
• Geometric phases in discrete dynamical systems
• Geometric Phases for Classical and Quantum Dynamics: Hannay angle and Berry Phase for Loops on a Torus