Controllability of the rolling system of a Lorentzian manifold on ${\mathbb R}^{n,1}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910563256238080 |
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| author | Osses, Abraham Bobadilla Molina, Mauricio Godoy |
| author_facet | Osses, Abraham Bobadilla Molina, Mauricio Godoy |
| contents | 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 $SO_0(n,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05863 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Controllability of the rolling system of a Lorentzian manifold on ${\mathbb R}^{n,1}$ Osses, Abraham Bobadilla Molina, Mauricio Godoy Differential Geometry Optimization and Control 53A17, 93B05, 53C50 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 $SO_0(n,1)$. |
| title | Controllability of the rolling system of a Lorentzian manifold on ${\mathbb R}^{n,1}$ |
| topic | Differential Geometry Optimization and Control 53A17, 93B05, 53C50 |
| url | https://arxiv.org/abs/2408.05863 |