On the bicycle transformation and the filament equation: results and conjectures

TL;DR

This paper explores the bicycle transformation in n-dimensional space, its relation to the filament equation, and provides evidence supporting its complete integrability, especially highlighting its bi-Hamiltonian structure in three dimensions.

Contribution

It introduces the bicycle correspondence as a model for bicycle kinematics, conjectures its complete integrability, and links it to the filament equation and bi-Hamiltonian structures.

Findings

Bicycle correspondence is conjectured to be completely integrable.

In 3D, it acts as a Bäcklund transformation for the filament equation.

The paper discusses bi-Hamiltonian properties and integrals of the bicycle correspondence.

Abstract

The paper concerns a simple model of bicycle kinematics: a bicycle is represented by an oriented segment of constant length in n-dimensional space that can move in such a way that the velocity of its rear end is aligned with the segment (the rear wheel is fixed on the bicycle frame). Starting with a closed trajectory of the rear end, one obtains the two respective trajectories of the front end, corresponding to the opposite directions of motion. These two curves are said to be in the bicycle correspondence. Conjecturally, this correspondence is completely integrable; we present a number of results substantiating this conjecture. In dimension three, the bicycle correspondence is the Backlund transformation for the filament equation; we discuss bi-Hamiltonian features of the bicycle correspondence and its integrals.