Introduction 1 2 Integrability of left-invariant systems on Lie groups with 2-dimensional coadjoint orbits 3 3 Classification of left-invariant Riemannian and sub-Riemannian metrics on three dimensional Lie groups 6 4 Explicit description of left-invariant geodesic flows on non-semisimple 3-dimensional Lie groups 11 5 Example: the group G VII 0 " Ć Ep2q 14 References 17
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.
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