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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.15871 |
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| _version_ | 1866916373360279552 |
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| author | Romero-Hernández, Luis G. Manuel-Cabrera, Jaime Chan-López, Ramón E. Paulin-Fuentes, Jorge M. |
| author_facet | Romero-Hernández, Luis G. Manuel-Cabrera, Jaime Chan-López, Ramón E. Paulin-Fuentes, Jorge M. |
| contents | This work conducts a Hamilton-Jacobi analysis of classical dynamical systems with internal constraints. We examine four systems, all previously analyzed by David Brown: three with familiar components (point masses, springs, rods, ropes, and pulleys) and one chosen specifically for its detailed illustration of the Dirac-Bergmann algorithm's logical steps. Including this fourth system allows for a direct and insightful comparison with the Hamilton-Jacobi formalism, thereby deepening our understanding of both methods. To provide a thorough analysis, we classify the systems based on their constraints: non-involutive, involutive, and a combination of both. We then use generalized brackets to ensure the theory's integrability, systematically remove non-involutive constraints, and derive the equations of motion. This approach effectively showcases the Hamilton-Jacobi method's ability to handle complex constraint structures. Additionally, our study includes an analysis of a gauge system, highlighting the versatility and broad applicability of the Hamilton-Jacobi formalism. By comparing our results with those from the Dirac-Bergmann and Faddeev-Jackiw algorithms, we demonstrate that the Hamilton-Jacobi approach is simpler and more efficient in its mathematical operations and offers advantages in computational implementation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15871 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular lagrangians and the Hamilton-Jacobi formalism in classical mechanics Romero-Hernández, Luis G. Manuel-Cabrera, Jaime Chan-López, Ramón E. Paulin-Fuentes, Jorge M. General Relativity and Quantum Cosmology This work conducts a Hamilton-Jacobi analysis of classical dynamical systems with internal constraints. We examine four systems, all previously analyzed by David Brown: three with familiar components (point masses, springs, rods, ropes, and pulleys) and one chosen specifically for its detailed illustration of the Dirac-Bergmann algorithm's logical steps. Including this fourth system allows for a direct and insightful comparison with the Hamilton-Jacobi formalism, thereby deepening our understanding of both methods. To provide a thorough analysis, we classify the systems based on their constraints: non-involutive, involutive, and a combination of both. We then use generalized brackets to ensure the theory's integrability, systematically remove non-involutive constraints, and derive the equations of motion. This approach effectively showcases the Hamilton-Jacobi method's ability to handle complex constraint structures. Additionally, our study includes an analysis of a gauge system, highlighting the versatility and broad applicability of the Hamilton-Jacobi formalism. By comparing our results with those from the Dirac-Bergmann and Faddeev-Jackiw algorithms, we demonstrate that the Hamilton-Jacobi approach is simpler and more efficient in its mathematical operations and offers advantages in computational implementation. |
| title | Singular lagrangians and the Hamilton-Jacobi formalism in classical mechanics |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2408.15871 |