Event Series
Event Type
Seminar
Thursday, November 21, 2019 4:30 PM
Sergei Tabachnikov (Penn State)
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, the model is quite rich and connected with several areas of research.  Here is a sampler. 

1) The front wheel's track and the bicycle's initial position determine the motion; the monodromy sending the initial to the terminal position turns out to be a Möbius transformation, a remarkable fact that has geometrical and dynamical consequences. 

2) The rear wheel's track and a choice of the direction of motion determine the front track; reversing the direction yields another front track.  This gives a discrete-time dynamical system on the space of curves, which is completely integrable and is related to the filament (a.k.a. binormal, smoke ring, local induction) equation. 

3) Given the front and rear tracks of a bicycle, can we tell which way the bicycle went?  Usually yes, but sometimes no.  The full description is an open problem, related to Ulam's problem: (in dimension 2) is the round ball the only body that floats in equilibrium in all attitudes?  This in turn is related to the motion of a charge in a magnetic field of a special kind.  The known solutions are solitons of the planar version of the filament equation.