In this thesis, we introduce a generalized class of billiard problems having their origin in rigid body motion. It is demonstrated that the free motion of any two or three dimensional rigid body colliding elastically between two flat, parallel walls is equivalent to a billiard system. In the two dimensional case, a broad class of traditional billiards (rectilinear point particle motion with specular reflections at the boundary walls) exhibiting integrable, near integrable (KAM), and chaotic motion of increasing randomness (K-flows, C-flows, and Bernoulli flows) is found. The specific case studied in some detail is that of a stick of zero width, which in turn is used to model the tossing of a coin. Furthermore, it is demonstrated that coin tossing, the prototypical example of an independent random process is a completely chaotic (Bernoulli) problem. This comes from the fact that the corresponding billiard system possesses strictly convex walls which is known to lead to Bernoulli type motion. Other evidence which indicates that coin tossing is (at least) a K-type process comes from an explicit Poincare surface of section map of the phase flow of this system and also from a calculation of the head to head correlation function.;The extension to three dimensional rigid body motion leads to a class of either non-Euclidean type billiards, interacting billiards, or both. The introduction of this curvilinear type motion is clearly related to the fact that the rotation group in three dimensions is nonabelian. It is further shown that the structure of the billiard walls is constructed solely from the geometric shape of the rigid body whereas the interacting field and the geodesic nature of the configuration space are determined by the inertial properties of the rigid body in question. Surprisingly, the curvilinear motion does not lead to a substantial decrease of stable periodic or quasi-periodic orbits normally found in the analogous rectilinear problem.;Finally, the free motion of a rigid one-dimensional stick colliding elastically within an infinitely massive circular wall is first shown to be equivalent to the three-dimensional motion of a billiard ball within a spiral column and then mapped onto a two-dimensional billiard problem with a rotating billiard wall. Indications that such a system has chaotic orbits and can possess integrable orbits is provided through the use of projected Poincare sections. When chaotic and integrable orbits co-exist, the chaotic trajectories appear in the form of Arnold's web. We also consider the limit of a stick of zero length in which the system becomes integrable.
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