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https://hdl.handle.net/2142/21890
Description
Title
Classical and quantum elliptic billiards
Author(s)
Crespi, Bruno
Issue Date
1990
Doctoral Committee Chair(s)
Chang, Shau-Jin
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Physics, General
Language
eng
Abstract
In classical mechanics we divide Hamiltonian systems into integrable and nonintegrable systems. This distinction corresponds to two different kinds of dynamical behavior. The motion of an integrable system is regular and predictable, whereas the time evolution of a nonintegrable system is, in certain regions of the phase space, chaotic and very sensitive to the change of initial conditions.
A Hamiltonian system, with n degrees of freedom, is integrable if it has n constants of motion in involution. This dynamical requirement is related to the topology of the flow in the phase space. The orbits of an integrable system are confined on n-dimensional surfaces which are topologically equivalent to an n-torus (if the flow is not singular).
Integrable systems are very rare (they have zero measure in the space of Hamiltonian systems), but they have a fundamental role in physics. The importance of integrable systems can be attributed in part to the fact that only these systems can be solved in a complete analytical form. The integrability of a physical system is usually the reflection of the existence of underlying symmetries and mathematical structures. Furthermore, in many cases, physically relevant systems can be treated as small perturbations of integrable models. As KAM theory explains, integrable properties play a determinant role in the dynamics of near-integrable systems.
"In the first part of this thesis we present a study of a class of integrable systems, the elliptic billiard and its extension to higher dimensions. We will show how the ""integrable"" topology of the phase space of these systems is related with the structure of the branch points in the corresponding Hamilton-Jacobi equations, and with the geometrical properties of the periodic orbits."
The principal result of this part is an algebraic derivation of Poncelet's theorem in three and higher dimensions. This study could be good starting point for more general investigation of the relation between integrability and the analytic properties of the classical equations.
In the second chapter we study how a distorsion of the boundary changes the quantum mechanics of the elliptic billiard, and we compare the modifications with the presence of chaotic orbits in the classical phase space. The purpose of this investigation is to see how the quantum mechanics of an integrable system responds to a nonintegrable perturbation.
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