  The space of $k$-jets of $n$ real function of one real variable $x$ admits
the structure of a Carnot group, which then has an associated Hamiltonian
geodesic flow. As in any Hamiltonian flow, a natural question is the existence
of periodic solutions. Does the space of $k$-jets have periodic geodesics? This
study will demonstrate the integrability of subRiemannian geodesic flow,
characterize and classify the subRiemannian geodesics in the space of $k$-jets,
and show that they are never periodic.
