This article contains a parametric formulation of the mechanics of rheonome systems. Its goal is to understand and explicitly express the influence of the nonstationancy of the bonds on the movement of the system. It is based on the family of possible varied trajectories, drawn from their initial position and on the transition to a new parameter, which depends on the chosen trajectory. Starting from the fundamental equation of dynamics, we have shown how in this way we can obtain the corresponding Alembert-Lagrange principle and pass from it to the general principle of Hamilton, which becomes variational under certain conditions. On this basis we have shown how we can formulate the proper Hamiltonian formalism, with an extended Hamiltonian, which is always possible here. The results obtained are in agreement with the analitic formalism modified by V. Vujičić (1980), in which a time function is chosen as an additional generalized coordinate, and a so-called rheonomic potential is introduced to the energy relations.