Date of Award

Winter 12-15-2022

Author's School

Graduate School of Arts and Sciences

Author's Department

Mathematics

Degree Name

Doctor of Philosophy (PhD)

Degree Type

Dissertation

Abstract

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive collision forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. Previous work indicated that no-slip billiards resemble non-holonomic systems, specifically, systems consisting of a ball rolling on surface. In prior research, such connections were only observed numerically and were restricted to very special surfaces. In this thesis, it is shown that no-slip billiard and rolling systems are directly related to each other under very general conditions. Our main result shows that no-slip billiards are truly the non-holonomic counterpart to standard billiard systems. In addition, to the best of our knowledge, we use a novel from of the rolling equations, showing that these systems are a one-parameter perturbation of the geodesic equation on a Riemannian manifold. This opens up a new area of investigation in the theory of geometric dynamical systems, concerning what we call rolling flows. We introduced the main concepts related to the rolling flow but we leave further development for future research.

Language

English (en)

Chair and Committee

Renato Feres

Committee Members

Timothy Chumley

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