Dynamics of bouncing rigid bodies and billiards in the spaces of constant curvature

Dynamical systems;billiards;horseshoe;adiabatic invariant;coin toss;periodic orbits

Kim, Ki Yeun

关键词: Dynamical systems;    billiards;    horseshoe;    adiabatic invariant;    coin toss;    periodic orbits;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/90586/KIM-DISSERTATION-2016.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 semi-dispersing boundary. In the first part of this dissertation, we study the collisions of a two-dimensional 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 equal-mass 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 three-period orbits in billiards on the Euclidean plane, the hyperbolic plane and on the two-dimensional sphere. We find that the set of three-period orbits in billiards on the hyperbolic plane has zero measure. For the sphere, three-period orbits can form a set of positive measure if and only if a certain natural condition on the orbit length is satisfied.