Separatrix crossing in rotation of a body with changing geometry of masses

TL;DR

This paper studies how a rotating body with changing mass distribution can cross the separatrix in its phase space, leading to unpredictable changes in its rotation, using adiabatic invariants and probabilistic analysis.

Contribution

It introduces a method to analyze separatrix crossings in a rotating body with slowly changing geometry, providing formulas for transition probabilities.

Findings

Crossing the separatrix causes quasi-random scattering in rotation dynamics.

Derived formulas for probabilities of different phase space domain captures.

Adiabatic invariants help describe the evolution of the body's rotation.

Abstract

We consider free rotation of a body whose parts move slowly with respect to each other under the action of internal forces. This problem can be considered as a perturbation of the Euler-Poinsot problem. The dynamics has an approximate conservation law - an adiabatic invariant. This allows to describe the evolution of rotation in the adiabatic approximation. The evolution leads to an overturn in the rotation of the body: the vector of angular velocity crosses the separatrix of the Euler-Poinsot problem. This crossing leads to a quasi-random scattering in body's dynamics. We obtain formulas for probabilities of capture into different domains in the phase space at separatrix crossings.