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

JOURNAL OF GEOMETRY AND PHYSICS

卷:115

Article

Tabachnikov, Serge¹ 

[1] Penn State Univ, Dept Math, University Pk, PA 16802 USA

关键词: Bicycle correspondence;    Monodromy;    Filament equation;    Complete integrability;    Backlund transformation;   

DOI  :  10.1016/j.geomphys.2016.05.013

来源: Elsevier

【 摘 要 】

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. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free