Constrained Dynamics on Eccentric Conic Orbits: Dirac-Bergmann and Hamilton-Jacobi Approaches

Alejandro G. Andarcia-Caballero; Jaime Manuel-Cabrera; Luis G. Romero-Hernández; Jorge M. Paulin-Fuentes

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arXiv · 2507.23271

Constrained Dynamics on Eccentric Conic Orbits: Dirac-Bergmann and Hamilton-Jacobi Approaches

Alejandro G. Andarcia-Caballero·Jaime Manuel-Cabrera·Luis G. Romero-Hernández·Jorge M. Paulin-Fuentes

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Abstract

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.

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Alejandro G. Andarcia-Caballero, Jaime Manuel-Cabrera, Luis G. Romero-Hernández, Jorge M. Paulin-Fuentes. 2025-07-31. Constrained Dynamics on Eccentric Conic Orbits: Dirac-Bergmann and Hamilton-Jacobi Approaches. https://arxiv.org/abs/2507.23271

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