Chaotic subRiemannian geodesic flow in $J^2(\mathbb{R}^2,\mathbb{R})$

Alejandro Bravo-Doddoli

Published 2022-07-20Version 1

The $2$-jets of space $J^2(\mathbb{R}^2,\mathbb{R})$ of a real function of one real variables $x$ and $y$ admits the structure of a Carnot group of step 3, as any subRiemannian manifold, $J^2(\mathbb{R}^2,\mathbb{R})$ has an associated Hamiltonian geodesic flow; A bijection between the geodesics and the $F$-curve is provided and used to prove that subRiemannian geodesic flow is not integrable. The later where, the $F$-curve is defined by a 2-degree of freedom Hamiltonian system with polynomial potential on $x$ and $y$.