No Periodic Geodesics in Jet Space

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 $T^{\ast }{J}^k$ for the geodesic flow and demonstrate that geodesics in $J^k$ are never periodic.

References in corpus (2)

• Geodesics in Jet Space
• Higher Elastica: Geodesics in the Jet Space