[Paper Review] Tire tracks and integrable curve evolution

Gil Bor, Mark Levi+2 more|arXiv (Cornell University)|May 17, 2017

Nonlinear Waves and Solitons34 references3 citations

TL;DR

This paper establishes deep connections between bicycle motion, integrable systems, and differential geometry by analyzing the bicycle equation—a first-order ODE governing rear wheel trajectories from a prescribed front track. It demonstrates that the bicycle monodromy in 3D encodes a Berry phase, shows that long bicycles act as approximate planimeters measuring area bivectors, and links bicycle correspondence to Darboux transformations and the AKNS/vortex filament systems, revealing that ambiguous closed curves are solitons of the planar filament equation.

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.

Motivation & Objective

• To understand the geometric and dynamical properties of the bicycle equation in n-dimensional Euclidean space.
• To establish connections between bicycle motion and integrable systems, particularly the AKNS hierarchy and the vortex filament equation.
• To analyze the monodromy map of the bicycle system and show its relation to the Berry phase in 3D.
• To demonstrate that long bicycles function as approximate planimeters measuring the area bivector of a closed front track.
• To characterize closed bicycle curves admitting multiple front tracks (bicycle correspondence) and relate them to solitons of the planar filament equation.

Proposed method

• Formulating the bicycle equation as a first-order differential equation that evolves the rear wheel trajectory from a given front track.
• Deriving the bicycle monodromy as a map on the sphere of initial orientations, shown to be a Möbius transformation in 2D and generalizable to higher dimensions.
• Using the Möbius group and Lie algebra isomorphisms (e.g., so_{2,1} ≅ sl_2(R)) to analyze the monodromy in 2D and 3D.
• Relating the bicycle system to the AKNS system via Darboux transformations, showing that bicycle correspondence is a special case of such transformations.
• Connecting the vortex filament equation to a 3D bicycle of imaginary length through a generating equation of the filament hierarchy.
• Applying Lyapunov-type arguments and asymptotic analysis to prove convergence of solutions as the bicycle length ℓ → 0, establishing smooth dependence on parameters.

Experimental results

Research questions

• RQ1 How does the bicycle monodromy in 3D relate to the Berry phase formula?
• RQ2 In what sense does a long bicycle approximate a planimeter measuring the area bivector of a closed front track?
• RQ3 What is the connection between bicycle correspondence and the Darboux transformation in the AKNS system?
• RQ4 How are ambiguous closed bicycle curves—admitting multiple front tracks—related to solitons of the planar filament equation?
• RQ5 What is the asymptotic behavior of the solution to the bicycle equation as the bicycle length ℓ approaches zero?

Key findings

• The bicycle monodromy in 3D is equivalent to a Berry phase formula, linking geometric holonomy to the evolution of the bicycle's orientation.
• A sufficiently long bicycle measures the area bivector of a closed front track with an error of order O(1/ℓ), confirming its role as a planimeter.
• Bicycle correspondence of space curves corresponds precisely to a Darboux transformation associated with the AKNS system.
• The 3D bicycle with imaginary length is governed by the vortex filament equation, and the entire filament hierarchy is encoded in a single generating equation.
• Ambiguous closed bicycle curves, such as those found by Wegner, are identified as buckled rings or solitons of the planar filament equation.
• Solutions to the bicycle equation converge smoothly to zero as the bicycle length ℓ → 0, and higher-order derivatives also converge, confirming analytic dependence on ℓ.