Geometry of Integrable Linkages
Dynamical Systems
2023-07-27 v1 Exactly Solvable and Integrable Systems
Abstract
In analogy with the well-known 2-linkage tractor-trailer problem, we define a 2-linkage problem in the plane with novel non-holonomic ``no-slip'' conditions. Using constructs from sub-Riemannian geometry, we look for geodesics corresponding to linkage motion with these constraints (``tricycle kinematics''). The paths of the three vertices turn out to be critical points for functionals which appear in the hierarchy of conserved quantities for the planar filament equation, a well known completely integrable evolution equation for planar curves. We show that the geodesic equations are completely integrable, and present a second connection to the planar filament equation.
Cite
@article{arxiv.2307.13839,
title = {Geometry of Integrable Linkages},
author = {Ron Perline and Sergei Tabachnikov},
journal= {arXiv preprint arXiv:2307.13839},
year = {2023}
}