Bicycling geodesics are Kirchhoff rods

Abstract

A bicycle path is a pair of trajectories in ${\mathbb R}^n$, the `front' and `back' tracks, traced out by the endpoints of a moving line segment of fixed length (the `bicycle frame') and tangent to the back track. Bicycle geodesics are bicycle paths whose front track's length is critical among all bicycle paths connecting two given placements of the line segment. We write down and study the associated variational equations, showing that for $n\geq 3$ each such geodesic is contained in a 3-dimensional affine subspace and that the front tracks of these geodesics form a certain subfamily of Kirchhoff rods, a class of curves introduced in 1859 by G. Kirchhoff, generalizing the planar elastic curves of J. Bernoulli and L. Euler.