Sub-Finsler geometry and nonholonomic mechanics
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916838742425600 |
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| author | Alabdulsada, Layth M. |
| author_facet | Alabdulsada, Layth M. |
| contents | We discuss a variational approach to the length functional and its relation to sub-Hamiltonian equations on sub-Finsler manifolds. Then, we introduce the notion of the nonholonomic sub-Finslerian structure and prove that the distributions are geodesically invariant concerning the Barthel non-linear connection. We provide necessary and sufficient conditions for the existence of the curves that are abnormal extremals; likewise, we provide necessary and sufficient conditions for normal extremals to be the motion of a free nonholonomic mechanical system, and vice versa. Moreover, we show that a coordinate-free approach for a free particle is a comparison between the solutions of the nonholonomic mechanical problem and the solutions of the Vakonomic dynamical problem for the nonholonomic sub-Finslerian structure. In addition, we provide an example of the nonholonomic sub-Finslerian structure. Finally, we show that the sub-Laplacian measures the curvature of the nonholonomic sub-Finslerian structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03489 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sub-Finsler geometry and nonholonomic mechanics Alabdulsada, Layth M. Differential Geometry Mathematical Physics 53C05, 53C60, 70F25, 53C17 We discuss a variational approach to the length functional and its relation to sub-Hamiltonian equations on sub-Finsler manifolds. Then, we introduce the notion of the nonholonomic sub-Finslerian structure and prove that the distributions are geodesically invariant concerning the Barthel non-linear connection. We provide necessary and sufficient conditions for the existence of the curves that are abnormal extremals; likewise, we provide necessary and sufficient conditions for normal extremals to be the motion of a free nonholonomic mechanical system, and vice versa. Moreover, we show that a coordinate-free approach for a free particle is a comparison between the solutions of the nonholonomic mechanical problem and the solutions of the Vakonomic dynamical problem for the nonholonomic sub-Finslerian structure. In addition, we provide an example of the nonholonomic sub-Finslerian structure. Finally, we show that the sub-Laplacian measures the curvature of the nonholonomic sub-Finslerian structure. |
| title | Sub-Finsler geometry and nonholonomic mechanics |
| topic | Differential Geometry Mathematical Physics 53C05, 53C60, 70F25, 53C17 |
| url | https://arxiv.org/abs/2309.03489 |