Geometry and Dynamics Seminar

Sergei Tabachnikov

(Pennsylvania State University)

"Flavors of "bicycle mathematics": tire tracks monodromy, hatchet planimeter, Menzin's conjecture, and oscillation of unicycle tracks"

Date: Tue, May 26, 2009
Time: 14:00
Place: Research I Seminar Room

Abstract: The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel, fixed on the bicycle frame); the same model describes the hatchet planimeter. In other words, we consider a non-holonomic system given by the standard contact structure on the space of contact elements. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and hence its terminal position; the monodromy map sends the initial position to the terminal one. This circle mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. Moebius transformations are either elliptic, or parabolic, or hyperbolic. I shall prove a 100 years old conjecture: if the front wheel track is an oval with area at least Pi then the respective monodromy is hyperbolic. I shall also discuss bicycle motions such that the rear wheel follows the trajectory of the front one. I shall explain why such "unicycle" tracks become more and more oscillating in forward direction and cannot be infinitely extended backward.