Wisdom System: Dynamics in the Adiabatic Approximation.

Abstract:

The Wisdom system is the Hamiltonian system with two degrees of freedom constructed as a simple approximate model describing some properties of the asteroid motion over long time scales. Its Hamiltonian is the principal part of the Hamiltonian of the planar elliptic restricted three body problem, averaged under the assumption of 3:1 mean-motion resonance. The phase variables of the Wisdom system evolve at different rates and can be subdivided into the 'fast' and 'slow' ones. Important feature of this system is the existence of an approximate integral, adiabatic invariant. We present detailed classification of the slow variables' evolution paths. We also consider properties of the adiabatic chaos area arising in the system's phase space. It emerges as a result of the adiabatic invariance violations generated by qualitative changes in the behaviour of fast variables. In particular, numerous stable long-periodic solutions are revealed.