Tire Tracks and Integrable Curve Evolution

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 AKNS (Ablowitz, Kaup, Newell and Segur) 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 3-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.

Special functions associated with automorphisms of the space of solutions to special double confluent Heun equation

“…Let pqrs-function obey the conditions of both Theorem 1 and Theorem 2, i.e. they are holomorphic on a connected and simply connected open set containing `i and `1 and meet the constraints (5), (11), and (12). Then the equalities M 1{2 rps " e iℓπ pλ `µ2 q…”

A family of integrable transformations of centroaffine polygons: geometrical aspects

“…For comparison, let M be the monodromy of the twisted triangle. Using formula (6), we find that Tr M = I.…”

“…The first one is the study of the discrete bicycle correspondence on polygons in the Euclidean plane [13], a discretization of the bicycle correspondence on smooth curves that was studied in [6,15]. See also [7,12] for the discrete and [18] for the continuous versions of this correspondence.…”

Locating the Sets of Exceptional Points in Dissipative Systems and the Self-Stability of Bicycles

“…The Whipple bike is a system consisting of four rigid bodies with knife-edge wheels making it non-holonomic, i.e., requiring for its description more configuration coordinates than the number of its admissible velocities [ 22 , 23 ]. Due to the non-holonomic constraints, even the bicycle tire tracks have a non-trivial and beautiful geometry that has deep and unexpected links to integrable systems, particle traps and the Berry phase [ 24 , 25 , 26 ].…”