Constraints and symmetry in mechanics of affine motion

TL;DR

This paper provides a geometric analysis of affine motion constraints in rigid body dynamics, comparing different models of rotation-less motion and their mathematical properties.

Contribution

It offers a detailed geometric interpretation of affine motion decompositions and examines various non-holonomic models of rotation-less motion, highlighting their differences and relationships.

Findings

Different equations for rotation-less motion models are derived.

The holonomic and non-holonomic models exhibit distinct mathematical properties.

Relationships between non-holonomic models and polar decomposition are identified.

Abstract

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.