Affine models of internal degrees of freedom and their action-angle description

We discuss equations of motion of small affinely-rigid bodies in the action-angle description. By “small” we mean a body of infinitesimal size moving in a Riemann space. Roughly speaking the extended body is replaced by the structured material point with internal degrees of freedom. We discuss some special cases of constant curvature two-dimensional spaces, like sphere and pseudosphere. Besides, motion on the torus manifold is discussed.