Dynamics Of One-dimensional Inelastic Particle Systems

In this thesis, we investigate the dynamics of one-dimensional inelastic particlesystems composed of rigid, frictionless, inelastic particles. Collisions between parti-cles are assumed to be inelastic with constant coefficient of restitution, and betweencollisions the particles move with constant velocity. We consider two different modelsand in each case consider the dynamics of an arbitrary number of particles of arbitrarymass. First, we consider a system bounded by two walls, with external forcing fromone of the walls. Second, we consider a system with periodic boundary conditions, thatcan also be thought of as a set of particles on a ring. We show that both systems exhibitsurprising behavior that is completely absent in equivalent elastic systems.In the first case, we investigate continuous transitions between different periodicorbits. We show that continuous transitions that occur when adding or subtracting asingle collision are, generically, of co-dimension 2. We give a full mechanical descrip-tion of the system and explain why this is the case. Surprisingly, we also show thatthere are an infinite set of degenerate transitions of co-dimension 1. We provide a the-oretical analysis that gives a simple criteria to classify which transitions are degeneratepurely using the discrete set of collisions that occur in the orbits. Our analysis allowsus to understand the nature of the degeneracy. We also show that higher degrees ofdegeneracy can occur, and provide an explanation.In the second case, we consider the dynamics of sequences of collisions that areself-similar in the sense that the relative positions return to their original relative posi-tions after the collision sequence, while the relative velocities are reduced by a constantfactor. For a given collision sequence, we develop the analytic machinery to determinethe particle velocities and the locations of collisions, and show that the problem ofdetermining self-similar orbits reduces to solving an eigenvalue problem to obtain theparticle velocities and solving a linear system to obtain the locations of inter-particlecollisions. For inelastic systems, we show that the collision locations can always beuniquely determined. We also show that this is in sharp contrast to the case of elastic systems in which infinite families of self-similar orbits can co-exist.