Higher elastica: geodesics in jet space

Abstract

Carnot groups are subRiemannian manifolds. As such, they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable; some are not. The k-jets space of for real-valued functions on the real line forms a Carnot group of dimension k+2. In this study, it is shown that its geodesic flow is integrable and that its geodesics generalize Euler’s elastica, with the case k=2 corresponding to the elastica.