Starting from the Fermat principle of geometrical optics, we analyse the ray dynamics in a graded
refractive index system device with cylindrical symmetry and a refractive index that decreases
parabolically with the radial coordinate. By applying Hamiltonian dynamics to the study of the ray
path we obtain the strict equivalence of this optical system with the dynamics of a particle with an
equivalent mass moving in a potential function that may exhibit a well, depending on the value of
some associated parameters. We analyse the features of this potential function as well as the energy
values and the symmetries of the system and see that both the azimuthal and axial components of the
optical conjugate momentum are two constants of motion. The phase space relation for the momentum
radial component is obtained analytically, and then we can obtain the components of the momentum
vector at any point, given the value of the radial coordinate, and from this we have the direction
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