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The space of $k$-jets of $n$ real function of one real variable $x$ admitsthe structure of a Carnot group, which then has an associated Hamiltoniangeodesic flow. As in any Hamiltonian flow, a natural question is the existenceof periodic solutions. Does the space of $k$-jets have periodic geodesics? Thisstudy 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.

No Periodic normal Geodesics in $J^k(\mathbb{R},\mathbb{R}^n)$