Problems and Techniques

Abstract

This installment of Problems and Techniques features articles on periodic billiard ball trajectories and the reduction of nearly quadratic polynomial Hamiltonians. Lorenz Halbeisen and Norbert Hungerbühler treat us, in our first article, to a study of periodic billiard ball trajectories on triangular tables. The problem of determining whether or not a billiard ball has a periodic path in a planar domain having a boundary composed of smooth curves has a long history with roots in the eighteenth century. G. D. Birkhoff, for example, established the existence of periodic trajectories on smooth convex domains in 1927. However, the case with nonsmooth boundaries is more difficult, and many situations on polygonal domains are still unresolved. The existence of periodic motion on acute triangles is perhaps the simplest case with motion following the orthoptic triangle with "bounces" at the bases of the three altitudes of the triangle. The existence of periodic trajectories on obtuse triangles is still undetermined and our authors describe cases where periodic motion can be established. They also examine the stability of such motion relative to small perturbations in the domain shape. I don't know if this analysis will improve your game, but it is indeed interesting mathematics. I hope that you enjoy it. In our second article, Jesús Palacián and Patricia Yanguas describe a procedure to reduce the number of degrees of freedom of two-dimensional Hamiltonians that are polynomial functions of the coordinates and momenta with a dominant part that is a homogeneous quadratic function. The popular Fermi--Pasta--Ulam and truncated Toda lattice systems are three-particle systems that may be expressed in this form after a suitable transformation of variables. The Hénon and Heiles family of systems also have the appropriate form. In these circumstances, our authors show how to reduce the number of degrees of freedom of the Hamilitonian system by one. The reduced Hamiltonians are then integrable and the reduced phase space is two-dimensional rather than four-dimensional. Thus, the phase-space diagrams are easily visualized, and this simplifies understanding. The authors characterize the reduced Hamiltonian systems in terms of a collection of normal forms and present conclusions regarding the perturbed Hamiltonian that result from properties of the reduced system. Lots of interesting views of the phase spaces for several reduced Hamiltonians are shown.