From point particles to body points

Given a spacetime background against which to observe it, a material system in motion can be modeled discretely, as a collection of particles called 'point masses', or continuously, as a dense and deformable object (a body, in the language of continuum mechanics) occupying a region consisting of 'material points'. In that the discrete and continuous descriptions have a common conceptual framework, the scale gap can be got over, provided a coarsening procedure is devised. Here we restrict attention to kinematics, and consider four such bottom-up procedures, two statistical and two deterministic, to derive a macroscopic velocity field from a microscopic one. We show that, to do this, introducing space and time mesoscopic scales is of the essence. We also show under what assumptions a macroscopic motion of the given material system can be associated with a microscopically-informed velocity field. Interestingly, no matter what coarsening procedure one chooses, the points of the resulting continuous description of the system's motion carry intrinsic physical information and may be persistently labelled: we propose to call them body points.