  The aim of this paper is to perform a deeper geometric analysis of problems
appearing in dynamics of affinely rigid bodies. First of all we present a
geometric interpretation of the polar and two-polar decomposition of affine
motion. Later on some additional constraints imposed on the affine motion are
reviewed, both holonomic and non-holonomic. In particular, we concentrate on
certain natural non-holonomic models of the rotation-less motion. We discuss
both the usual d'Alembert model and the vakonomic dynamics. The resulting
equations are quite different. It is not yet clear which model is practically
better. In any case they both are different from the holonomic constraints
defining the rotation-less motion as a time-dependent family of symmetric
matrices of placements. The latter model seems to be non-geometric and
non-physical. Nevertheless, there are certain relationships between our
non-holonomic models and the polar decomposition.
