PhD Thesis Defense: "Geometry and Dynamics of Rolling Systems"

Abstract: Billiard systems may be regarded generally as models of mechanical systems in which rigid parts interact through elastic impulsive forces. A type of billiard system called no-slip billiard has been used to account for linear/angular momentum exchange in collisions involving a spherical body, Based on an idea by Borisov, Kilin and Mamaev, we show that no-slip billiards very generally arise as limits of rolling systems, akin to how ordinary billiards can be expressed as limits of geodesic flows through a flattening of the Riemannian manifold. In chapter 2 we give a general setup of rolling systems. In chapter 3 we establish some definitions and properties for no-slip billiard. In Chapter 4 we review previous work on connection between the two types of systems. In chapter 5 we prove the main result that no-slip billiards arise from rolling systems by taking a limit. In Chapter 6 we look at future directions of establishing a more systematic dynamical theory for rolling systems on polygonal billiards, Lyapunov stability and the hyperbolic and elliptical properties of the rolling systems, in the spirit of geodesic flows.

Host: Renato Feres