This talk concerns a naive model of bicycle motion: a
bicycle is a segment of fixed length that can move so that the
velocity of the rear end is always aligned with the segment.
Surprisingly, this simple model is quite rich. Here is a sampler of
the problems that I’d like to discuss.
1) The trajectory of the front wheel and the initial position of the
bicycle uniquely determine its motion and its terminal position; the
monodromy map sending the initial position to the terminal one arises.
This circle mapping is a Moebius transformation, a remarkable fact
that has various geometrical and dynamical consequences. Moebius
transformations belong to one of the three types: elliptic, parabolic
and hyperbolic. I shall outline a proof of a 100 years old conjecture:
if the front wheel track of a unit length bicycle is an oval with area
at least Pi then the respective monodromy is hyperbolic.
2) The rear wheel track and a choice of the direction of motion
uniquely determine the front wheel track; changing the direction to
the opposite, yields another front track. These two front tracks are
related by the bicycle (Backlund, Darboux) correspondence which
defines a discrete time dynamical system on the space of curves. What
do pairs of curves in the bicycle correspondence have in common? It
turns out, infinitely many quantities (the perimeter length, the total
curvature squared,…) I shall explain that the bicycle correspondence
is closely related with another, well studied, completely integrable
dynamical system, the filament (a.k.a binormal, smoke ring, local
3) Given the rear and front tracks of a bicycle, can one tell which
way the bicycle went? Usually, one can, but sometimes one cannot. The
description of these ambiguous tire tracks is an open problem,
intimately related with Ulam's problem in flotation theory (in
dimension two): is the round ball the only body that floats in
equilibrium in all positions?