Controllability of the rolling system of a Lorentzian manifold on ${\mathbb{R}}^{n,1}$

Abstract

In this paper, we study the mechanical system associated with rolling a Lorentzian manifold $(M,g)$ of dimension $n+1\ge 2$ on flat Lorentzian space $\hat{M}={\mathbb{R}}^{n,1}$, without slipping or twisting. Using previous results, it is known that there exists a distribution ${\mathcal{D}}_R$ of rank $(n+1)$ defined on the configuration space $Q(M,\hat{M})$ of the rolling system, encoding the no-slip and no-twist conditions. Our objective is to study the problem of complete controllability of the control system associated with ${\mathcal{D}}_R$. The key lies in examining the holonomy group of the distribution ${\mathcal{D}}_R$ and, following the approach of \cite{ChKok}, establishing that the rolling problem is completely controllable if and only if the holonomy group of $(M,g)$ equals $S{O}_0(n,1)$.

13 pages