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

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.