Geometry of Integrable Linkages

TL;DR

This paper explores the geometry of a novel 2-linkage problem with non-holonomic constraints, revealing its integrability and connections to the planar filament equation through sub-Riemannian geometry.

Contribution

It introduces a new 2-linkage problem with no-slip conditions, demonstrating its complete integrability and linking it to the planar filament equation.

Findings

Geodesics are critical points of conserved functionals.

The geodesic equations are completely integrable.

Connections to the planar filament equation are established.

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.