Models of internal degrees of freedom based on classical groups and their homogeneous spaces

We derive and discuss equations of motion of an infinitesimal affinely-rigid body moving in a Riemannian space. The main novelty here is that the internal motion is described with respect to some prescribed field of orthonormal frames. This enables us to describe effectively interactions between internal rotations and homogeneous deformations. One can obtain smoothly the limiting rigid body case and discuss the problem of small elastic deformations imposed on rigid rotations. Some physically realistic and integrable models in two dimensions are discussed. We were motivated by the pure analytical mechanics and theory of dynamical systems, nevertheless some physical applications are not excluded, e.g. the motion of continental plates on the surface of the Earth.