Some generalised models in rigid body dynamics

This thesis contains two main chapters.

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

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.

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

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.