No Periodic Geodesics in Jet Space

TL;DR

This paper proves that the space of k-jets of real functions, equipped with a sub-Riemannian structure, has no periodic geodesics, by explicitly constructing action-angle coordinates and analyzing the flow.

Contribution

It provides the first explicit construction of action-angle coordinates for the geodesic flow on jet spaces and shows the non-existence of periodic geodesics in this setting.

Findings

Geodesic flow on jet spaces is integrable.

No periodic geodesics exist in $J^k$.

Explicit action-angle coordinates are constructed.

Abstract

The $J^k$ space of $k$-jets of a real function of one real variable $x$ admits the structure of a sub-Riemannian manifold, which then has an associated Hamiltonian geodesic flow, and it is integrable. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does $J^k$ have periodic geodesics? This study will find the action-angle coordinates in $T^*J^k$ for the geodesic flow and demonstrate that geodesics in $J^k$ are never periodic.