### 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.