Journée de Rham 2017 : Flavors of bicycle mathematics

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 plan 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 induction) equation. 

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? ​