Dancing polygons, rolling balls and the Cartan-Engel distribution
| Autor: | Bor, Gil, Lamoneda, Luis Hernández |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | A pair of planar polygons is "dancing" if one is inscribed in the other and they satisfy a certain cross-ratio relation at each vertex of the circumscribing polygon. Non-degenerate dancing pairs of closed $n$-gons exist for all $n\geq 6$. Dancing pairs correspond to trajectories of a non-holonomic mechanical system, consisting of a ball rolling, without slipping and twisting, along a polygon drawn on the surface of a ball 3 times larger than the rolling ball. The correspondence stems from reformulating both systems as piecewise rigid curves of a certain remarkable rank 2 non-integrable distribution defined on a 5-dimensional quadric in $\mathbb{RP}^6$, introduced by \'E. Cartan and F. Engel in 1893 in order to define the simple Lie group $\mathrm{G}_2$. Comment: 36 pages, 16 figures. Small corrections, remark 2 expanded. To appear in NY J of Math |
| Databáze: | arXiv |
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