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Title:  Dynamics of bouncing rigid bodies and billiards in the spaces of constant curvature 
Author(s):  Kim, Ki Yeun 
Director of Research:  Zharnitsky, Vadim 
Doctoral Committee Chair(s):  Baryshnikov, Yuliy 
Doctoral Committee Member(s):  DeVille, Robert; Rapti, Zoi 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Dynamical systems
billiards horseshoe adiabatic invariant coin toss periodic orbits 
Abstract:  Mathematical billiard is a dynamical system studying the motion of a mass point inside a domain. The point moves along a straight line in the domain and makes specular reflections at the boundary. The theory of billiards has developed extensively for itself and for further applications. For example, billiards serve as natural models to many systems involving elastic collisions. One notable example is the system of spherical gas particles, which can be described as a billiard on a higher dimensional space with a semidispersing boundary. In the first part of this dissertation, we study the collisions of a twodimensional rigid body using billiard dynamics. We first define a dumbbell system, which consists of two point masses connected by a weightless rod. We assume the dumbbell moves freely in the air and makes elastic collisions at a flat boundary. For arbitrary mass choices, we use billiard techniques to find the sharp bound on the number of collisions of the dumbbell system in terms of the mass ratio. In the limiting case where the mass ratio is large and the dumbbell rotates fast, we prove that the system has an adiabatic invariant. In case the two masses of the dumbbell are equal, we assume gravity in the system and study its infinitely many collisions. In particular, we analytically verify that a Smale horseshoe structure is embedded in the billiard map arising from the equalmass dumbbell system. The second part of this dissertation concerns the billiards in the spaces of constant curvature. We provide a unified proof to classify the sets of threeperiod orbits in billiards on the Euclidean plane, the hyperbolic plane and on the twodimensional sphere. We find that the set of threeperiod orbits in billiards on the hyperbolic plane has zero measure. For the sphere, threeperiod orbits can form a set of positive measure if and only if a certain natural condition on the orbit length is satisfied. 
Issue Date:  20160421 
Type:  Thesis 
URI:  http://hdl.handle.net/2142/90586 
Rights Information:  Copyright 2016 Ki Yeun Kim 
Date Available in IDEALS:  20160707 
Date Deposited:  201605 
This item appears in the following Collection(s)

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois