Conformally symplectic Chaplygin reduction in rubber rolling of surfaces of revolution over the plane
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| Format: | Preprint |
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2025
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| _version_ | 1866914114522054656 |
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| author | Koiller, Jair |
| author_facet | Koiller, Jair |
| contents | Rubber rolling (no-slip and no-twist) of a convex body on the plane under the influence of gravity is a SE(2) Chaplygin system, that reduces to the sphere of Poisson vectors. I comment upon an observation by A.V Borisov and I.S. Mamaev (Regular and Chaotic Dynamics, 13(5):443-490, 2008) for the case of surfaces of revolution [also in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (Regular and Chaotic Dynamics, 18(3):277-328, 2013)]. They show that this case is quite special: the additional integral of motion is elementary, while for marble rolling it is not elementary. I connect this finding with recent work about Chaplygin reduced systems that are conformally symplectic (Luis Garcia Naranjo and Juan C. Marrero. The geometry of nonholonomic Chaplygin systems revisited. Nonlinearity, 33(3):1297, 2020). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_22032 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conformally symplectic Chaplygin reduction in rubber rolling of surfaces of revolution over the plane Koiller, Jair Mathematical Physics 37J60, 70F25, 58A15, 58A30 Rubber rolling (no-slip and no-twist) of a convex body on the plane under the influence of gravity is a SE(2) Chaplygin system, that reduces to the sphere of Poisson vectors. I comment upon an observation by A.V Borisov and I.S. Mamaev (Regular and Chaotic Dynamics, 13(5):443-490, 2008) for the case of surfaces of revolution [also in A. V. Borisov, I. S. Mamaev and I. A. Bizyaev (Regular and Chaotic Dynamics, 18(3):277-328, 2013)]. They show that this case is quite special: the additional integral of motion is elementary, while for marble rolling it is not elementary. I connect this finding with recent work about Chaplygin reduced systems that are conformally symplectic (Luis Garcia Naranjo and Juan C. Marrero. The geometry of nonholonomic Chaplygin systems revisited. Nonlinearity, 33(3):1297, 2020). |
| title | Conformally symplectic Chaplygin reduction in rubber rolling of surfaces of revolution over the plane |
| topic | Mathematical Physics 37J60, 70F25, 58A15, 58A30 |
| url | https://arxiv.org/abs/2510.22032 |