Galilean invariance and the Lorentz transformation are the two pillars of mechanics that an equation of motion must respect. The objective is to extend the Galilean invariance to uniform rotational motion and to expansion motion of which motion at constant translational velocity is only a special case. The second goal is to show that the discrete equation of motion is naturally relativistic without using the Lorentz transformation. The realization of these two aims leads to different concepts of special relativity: (i) space is homogeneous and isotropic and (ii) time, which flows regularly, is invariant by translation. These invariances, symmetric for translation and rotation, observed by the discrete equation of motion give it fundamental conservation properties to extend the scope of the equations of fluid mechanics to other fields of physics.