Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917087112331264 |
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| author | Peng, Linyu Yoshimura, Hiroaki |
| author_facet | Peng, Linyu Yoshimura, Hiroaki |
| contents | In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the $(\pm)$-induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed in Cortés and Martínez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09530 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics Peng, Linyu Yoshimura, Hiroaki Mathematical Physics Numerical Analysis Differential Geometry Dynamical Systems 37J06, 37J39, 37J60, 37N30, 65P10, 70H15 G.2.0 In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the $(\pm)$-induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed in Cortés and Martínez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests. |
| title | Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics |
| topic | Mathematical Physics Numerical Analysis Differential Geometry Dynamical Systems 37J06, 37J39, 37J60, 37N30, 65P10, 70H15 G.2.0 |
| url | https://arxiv.org/abs/2411.09530 |