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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.23271 |
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| _version_ | 1866908473512427520 |
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| author | Andarcia-Caballero, Alejandro G. Manuel-Cabrera, Jaime Romero-Hernández, Luis G. Paulin-Fuentes, Jorge M. |
| author_facet | Andarcia-Caballero, Alejandro G. Manuel-Cabrera, Jaime Romero-Hernández, Luis G. Paulin-Fuentes, Jorge M. |
| contents | In this work, we investigate a Lagrangian model describing a particle constrained to move along non-degenerate conic sections, parameterized by the orbital eccentricity \( e \). In the non-relativistic regime, we apply the Dirac--Bergmann algorithm to identify a set of four second-class constraints, compute the corresponding Dirac brackets, and isolate the true physical degrees of freedom. This procedure yields a unified Hamiltonian treatment of circular (\( e = 0 \)), elliptical (\( 0 < e < 1 \)), parabolic (\( e = 1 \)), and hyperbolic (\( e > 1 \)) trajectories. We then extend the analysis to the relativistic case, where we observe a similar constraint structure and construct the associated Dirac brackets accordingly. Finally, using the Hamilton-Jacobi formalism, we identify a set of non-involutive constraints; by introducing generalized brackets, we restore integrability and derive the correct equations of motion. A comparative analysis of both formalisms highlights their complementary features and deepens our understanding of the dynamics governing particles restricted to conic geometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constrained Dynamics on Eccentric Conic Orbits: Dirac-Bergmann and Hamilton-Jacobi Approaches Andarcia-Caballero, Alejandro G. Manuel-Cabrera, Jaime Romero-Hernández, Luis G. Paulin-Fuentes, Jorge M. General Relativity and Quantum Cosmology In this work, we investigate a Lagrangian model describing a particle constrained to move along non-degenerate conic sections, parameterized by the orbital eccentricity \( e \). In the non-relativistic regime, we apply the Dirac--Bergmann algorithm to identify a set of four second-class constraints, compute the corresponding Dirac brackets, and isolate the true physical degrees of freedom. This procedure yields a unified Hamiltonian treatment of circular (\( e = 0 \)), elliptical (\( 0 < e < 1 \)), parabolic (\( e = 1 \)), and hyperbolic (\( e > 1 \)) trajectories. We then extend the analysis to the relativistic case, where we observe a similar constraint structure and construct the associated Dirac brackets accordingly. Finally, using the Hamilton-Jacobi formalism, we identify a set of non-involutive constraints; by introducing generalized brackets, we restore integrability and derive the correct equations of motion. A comparative analysis of both formalisms highlights their complementary features and deepens our understanding of the dynamics governing particles restricted to conic geometries. |
| title | Constrained Dynamics on Eccentric Conic Orbits: Dirac-Bergmann and Hamilton-Jacobi Approaches |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2507.23271 |