We propose the Hamiltonian description of the streamline formalism, which includes: (a) the canonical elements for the unperturbed epicyclic motion and (b) the perturbation equations when the streamlines are perturbed both by conservative and dissipative forces. As an application to planetary ring dynamics, we use the Hamiltonian description to derive a mapping system for the self-gravity model of elliptical rings. The canonical epicyclic elements are also useful in satellites dynamics when the oblateness of the central body is the main perturbation.