The simplest model of a bicycle is a segment of fixed length that can move, in n-dimensional Euclidean space, so that the velocity of the rear end is always aligned with the segment (the rear wheel is fixed on the frame). The rear wheel track and a choice of direction uniquely determine the front wheel track; changing the direction to the opposite yields another front track. The two tracks are related by the bicycle (Darboux, Backlund) transformation defining a discrete time dynamical system on the space of closed curves. This system is completely integrable. I shall discuss the symplectic, and in dimension 3, bi-symplectic, nature of this transformation and, also in dimension 3, its relation with the filament (binormal, smoke ring, local induction) equation. An interesting problem is to describe the curves that are in the bicycle correspondence with themselves (in this case, given the front and rear tracks, one cannot tell which way the bicycle went). In dimension two, such curves yield solutions to Ulam's problem: is the round ball the only body that floats in equilibrium in all positions? I shall discuss Franz Wegner's (a Heidelberg physicist) results on this problem and relate them with solitons of the planar filament equation.
Freitag, den 2. Juni 2017 um 16.00 Uhr, in Mathematikon, 5.104, 5th floor Freitag, den 2. Juni 2017 at 16.00, in Mathematikon, 5.104, 5th floor
Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Peter Albers