Periodic solutions of 3-beads on a ring and right-triangular billiards
Collisions of beads on a ring
Daniel Philip Newton
san marcos high school
Abstract
When three beads slide on a ring and collide with elastic collisions, if their velocities are chosen carefully, they undergo periodic motion. We compare this problem in mechanics to a geometric problem in billiard dynamics in a right-triangle. The billiard problem uses specular reflection at the boundaries (angle in equals angle out), whereas the boundary conditions for bead collisions follow from the conservation of energy and momentum which depend on the relative velocities of the beads. For the billiard problem, using a sequence of reflections of the right triangle, we highlight techniques to find special families of periodic orbits that are parallel to one of the boundaries, or oblique to the hypotenuse at certain discrete angles. For the bead collision problem, we show how this approach can be generalized to one in which the triangles are not only reflected across boundaries, but also rotated around the point of collision to adjust for the fact that the boundary condition is non-specular. All of the results described in the paper are accessible to high school or college level pre-calculus students since the techniques rely only on geometry, trigonometry, symmetries and reflections, algebra, simple number theory, and basic laws of mechanics.
