Sub-Riemannian geodesics on the free Carnot group with the growth vector (2,3,5,8)

Abstract: We consider the free nilpotent Lie algebra $L$ with 2 generators, of step 4, and the corresponding connected simply connected Lie group $G$. We study the left-invariant sub-Riemannian structure on $G$ defined by the generators of $L$ as an orthonormal frame. We compute two vector field models of $L$ by polynomial vector fields in $R^8$, and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in $G$, the right-invariant frame on $G$, linear on fibers Hamiltonians corresponding to the left-invariant and right-invariant frames on $G$, Casimir functions and co-adjoint orbits on $L^*$. Via Pontryagin maximum principle, we describe abnormal extremals and derive a Hamiltonian system $\dot \lambda = \vec{H}(\lambda)$, $\lambda \in T^*G$, for normal extremals. We compute 10 independent integrals of $\vec{H}$, of which only 7 are in involution. After reduction by 4 Casimir functions, the vertical subsystem of $\vec{H}$ on $L^*$ shows numerically a chaotic dynamics, which leads to a conjecture on non-integrability of $\vec{H}$ in the Liouville sense.