Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater

Abstract

One of the main approaches to investigating sub-Riemannian problems is Mitchell's theorem on nilpotent approximation, which reduces the analysis of a neighbourhood of a regular point to the analysis of the left-invariant sub-Riemannian problem on the corresponding Carnot group. Usually, the in-depth investigation of sub-Riemannian shortest paths is based on integrating the Hamiltonian system of Pontryagin's maximum principle explicitly. We give new formulae for sub-Riemannian geodesics on a Carnot group with growth vector and prove that left-invariant sub-Riemannian problems on free Carnot groups of step 4 or greater are Liouville nonintegrable. Bibliography: 30 titles.