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

TL;DR

This paper investigates the controllability of a mechanical system involving the rolling of a Lorentzian manifold on flat Lorentzian space, establishing that controllability depends on the holonomy group being the Lorentz group.

Contribution

It proves that the rolling system is completely controllable if and only if the manifold's holonomy group is the Lorentz group SO_0(n,1).

Findings

Controllability is characterized by the holonomy group.

The system is controllable if holonomy equals SO_0(n,1).

Holonomy analysis determines the controllability of the rolling system.

Abstract

In this paper, we study the mechanical system associated with rolling a Lorentzian manifold $(M,g)$ of dimension $n+1\geq2$ on flat Lorentzian space $\widehat{M}={\mathbb R}^{n,1}$, 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 $Q(M,\widehat{M})$ 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 $SO_0(n,1)$.