Iterated Impact Dynamics of N-Beads on a Ring

Abstract

When N-beads slide along a frictionless hoop, their collision sequence gives rise to a dynamical system that can be studied via matrix products. It is of general interest to understand the distribution of velocities and the corresponding eigenvalue spectrum that a given collision sequence can produce. We formulate the problem for general N and state some basic theorems regarding the eigenvalues of the collision matrices and their products. The case of three beads of masses m1, m2, m3 is studied in detail. We exploit the fact that each collision sequence can be viewed as a billiard trajectory in a right triangle with non-standard reflection rules. Existence of families of periodic orbits are proven, and orbits that densely fill the triangle are computed. Eigenvalue distributions and position and velocity histograms are computed as a function of the restitution coefficient, both periodic and dense collision sequences are discussed, and a series of conjectures based on computational evidence are formulated. Comparisons are made between the eigenvalue distributions and autocorrelation matrices associated with dense trajectories generated from a chaotic collision sequence and spectra from matrix sequences generated from random orderings, and we describe how the three-bead system could be used as the basis for a random number generating algorithm that is computationally efficient.