Higher elastica: geodesics in jet space

Carnot groups are subRiemannian manifolds. As such they admit geodesic flows, which left-invariant Hamiltonian flows on their cotangent bundles. Some of these integrable. not. The space k-jets for real-valued functions the real line forms a group dimension $k+2$. We show that its flow is integrable and geodesics generalize Euler's elastica, with case $k=2$ corresponding to as shown by Sachkov Ardentov.