Some aspects of affine motion and nonholonomic constraints. Two ways to describe homogeneously deformable bodies

TL;DR

This paper analyzes the geometric dynamics of affinely-rigid bodies using classical and variational methods, revealing significant differences and highlighting the elegance of the vakonomic approach.

Contribution

It compares classical d'Alembert and vakonomic descriptions of affinely-rigid bodies, providing insights into their differences and mathematical properties.

Findings

Vakonomic model appears more elegant mathematically.

Classical and vakonomic methods yield different results.

Analysis enhances understanding of affine motion and nonholonomic constraints.

Abstract

This paper has been inspired by ideas presented by V. V. Kozlov in his works [19, 20]. In this paper our goal is to carry out a thorough analysis of some geometric problems of the dynamics of affinely-rigid bodies. We present two ways to describe this case: the classical dynamical d'Alembert and variational, i.e., vakonomic one. So far, we can see that they give quite different results. The vakonomic model from the mathematical point of view seems to be more elegant. The similar problems were examined by J\`o\'zwikowski and W. Respondek in their paper [16]