| dc.description.abstract | One of the prime paradigms of nonlinear dynamics, billiard systems offer a simple setting to examine regular and chaotic motion. Gravitational billiards are generalizations of classical billiards ideal for both analytical and experimental investigations. In this thesis, we study the nonlinear dynamics of two natural generalizations of one of the most widely studied Hamiltonian gravitational billiards, the wedge billiard. First, we incorporate time-dependence into the system through sinusoidal driving of the wedge. We introduce a model describing the driven wedge in terms of a four-dimensional discrete map, and analyze the properties of this map analytically and numerically. Unbounded orbits in the form of Fermi acceleration are confirmed for elastic collisions, and regular and chaotic attractors are found for inelastic collisions. Next, we examine the natural three-dimensional generalization of the wedge billiard: the conic billiard. Namely, we consider the motion of a classical particle in a constant gravitational field, colliding elastically with a linear cone of vertex angle 2 theta. We derive a two-dimensional area-preserving map characterizing the dynamics, and demonstrate several integrable limits of the system. We compute some simple periodic orbits and analyze their stability as a function of parameters, and present some additional numerical results. We find that for small values of L, the z-component of angular momentum, the conic billiard exhibits behavior characteristic of two-degree-of-freedom Hamiltonian systems with a discontinuity, and the dynamics are qualitatively similar to that of the wedge billiard. As we increase L the dynamics become less chaotic, and the correspondence with the wedge billiard is lost. | |
