Relativistic conic motion as a second-class dynamical system

We investigate relativistic motion along a general conic path under the influence of an open potential as a Dirac-Bergmann constrained dynamical model. The system turns out to exhibit a set of four second-class constraints in phase space which we fully explore obtaining a relativistic Poisson algebra generalizing previously known algebraic structures. With a convenient integration factor, the Euler-Lagrange differential equations can be worked out to its general solution in closed form. We perform the canonical quantization in terms of the corresponding Dirac brackets, applying the Dirac-Bergmann algorithm. The complete Dirac brackets algebra in phase space as well as its physical realization in terms of differential operators are explicitly obtained.