Tire Tracks and Integrable Curve Evolution

Abstract

Abstract We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation—the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the Ablowitz, Kaup, Newell, and Segur (AKNS) system and the vortex filament equation. We show that “bicycle correspondence” of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a three-dimensional bike of imaginary length. We show that a series of examples of “ambiguous” closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.