A new perspective on nonholonomic brackets and Hamilton-Jacobi theory
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909090485108736 |
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| author | de León, Manuel Lainz, Manuel López-Gordón, Asier Marrero, Juan Carlos |
| author_facet | de León, Manuel Lainz, Manuel López-Gordón, Asier Marrero, Juan Carlos |
| contents | The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_06049 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new perspective on nonholonomic brackets and Hamilton-Jacobi theory de León, Manuel Lainz, Manuel López-Gordón, Asier Marrero, Juan Carlos Mathematical Physics Differential Geometry Dynamical Systems primary: 37J60, 70F25, 70H20, secondary: 53D17, 53Z05, 70G45 The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems. |
| title | A new perspective on nonholonomic brackets and Hamilton-Jacobi theory |
| topic | Mathematical Physics Differential Geometry Dynamical Systems primary: 37J60, 70F25, 70H20, secondary: 53D17, 53Z05, 70G45 |
| url | https://arxiv.org/abs/2307.06049 |