Some generalised models in rigid body dynamics

<div>This thesis contains two main chapters.</div><div>In the first main chapter 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 us to describe the evolution of rotation in the adiabatic approximation. The evolution leads to an overturn in the rotation</div><div>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.</div><div>In the second main chapter we consider the Liouville integrability of left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensional Lie group G. In the case of G = SO(3) such systems coincide with the classical equations describing the motion of a free rigid body so that our studies can be naturally regarded as a generalisation of rigid body dynamics to the other Lie groups in dimension 3. We derive the integrability property from the fact that the coadjoint orbits of G are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension. We also give normal forms for left-invariant Riemannian and sub-Riemannian metrics on 3-dimensional Lie groups focusing on the case of solvable groups, as the cases of SO(3) and SL(2) have been already extensively studied. Our description is explicit and is given in</div><div>global coordinates on G which allows one to easily obtain parametric equations of geodesics in quadratures. Finally, we study generalised systems related to 3-dimensional Lie groups and ?find some series of Liouville integrable cases among them.</div>