Integrability and Chaotic Behavior in Mechanical Billiard Systems

This thesis is devoted to the study of mathematical billiards in the presence of non-constant potentials and their integrability and chaotic behavior.
Classical examples of integrable billiards are free billiards in circles and ellipses. In the presence of specific potentials (such as Kepler potential and harmonic (Hooke) potential), there are various known integrable billiard systems. These integrable examples have been found independently in different contexts. In Chapter 2, we illustrate how some of these integrable billiard systems are related to each other by conformal transformations. As an application, we obtain infinitely many billiard systems defined in central force problems which are integrable on a particular energy level. We then explain that the classical Hooke-Kepler correspondence extends to the correspondence between integrable Hooke and Kepler billiards. As a result, we show that any focused conic sections give rise to integrable Kepler billiards which give new examples of integrable Kepler billiards. The conformal transformation technique is applied to Stark-type problems and Euler's two-center problem and provides new examples of integrable mechanical billiards. ... mehr
In Chapter 3 we show that integrable Kepler and Hooke billiard systems on the plane have the corresponding integrable billiard systems on surfaces of constant curvatures. We also establish the integrability of a class of billiard systems defined in the Lagrangian problem, which is the superposition of two Kepler problems and a Hooke problem, on the sphere, in the plane, and in the hyperbolic plane. These results are obtained by the method of projective dynamics and projective billiards.
A toy model of billiard systems with a central force problem in the plane and with a line as the reflection wall was proposed by L. Boltzmann to illustrate his ergodic hypothesis. Later, it has been found that not all such systems are ergodic, and it becomes a question whether some of such systems are ergodic.
In Chapter 4, we compute the billiard mappings of Boltzmann's billiard systems, and we present some numerical studies on their chaotic behavior and ergodicity. We found some numerical evidence suggesting that some of these systems might be ergodic.