Symmetry and its applications in mechanics

Abstract

In this thesis, we observe the role of non-holonomic constraints imposed on the unit ball that rolls in the plane without slipping or turning. These constraints implicate a holonomy phenomenon. We want to determine the optimal motions of that rolling ball. The optimal rotations that describe the optimal motions satisfy a differential equation which is not directly integrable in general. Using holonomy, we obtain closed formulae for the optimal motions. In the last part of this thesis, we develop a numerical scheme to compute the optimal rotations numerically and to validate our closed form solutions.