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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2406.16336 |
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| _version_ | 1866909230160674816 |
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| author | Eckmann, Jean-Pierre Sobolev, Yaroslav I. Tlusty, Tsvi |
| author_facet | Eckmann, Jean-Pierre Sobolev, Yaroslav I. Tlusty, Tsvi |
| contents | A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane $\mathbb{R}^2$ can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve - most often two copies will suffice, but some curves require an arbitrarily large number of copies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16336 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tumbling Downhill along a Given Curve Eckmann, Jean-Pierre Sobolev, Yaroslav I. Tlusty, Tsvi Mathematical Physics Differential Geometry Classical Physics A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane $\mathbb{R}^2$ can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve - most often two copies will suffice, but some curves require an arbitrarily large number of copies. |
| title | Tumbling Downhill along a Given Curve |
| topic | Mathematical Physics Differential Geometry Classical Physics |
| url | https://arxiv.org/abs/2406.16336 |