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Known Examples of Material Points

The real difference between a material particle and a material point is therefore that this last one conceptually accommodates the ideas of variability and destructibility, which are essential attributes of the cosmic bodies which the material point is intended to represent. And we are in need of describing this variability and destructibility, as the last century Physics plainly shows. This desideratum can only be accomplished classically by considering that a material point can be modeled as an ensemble of material particles. As the idea of ensemble immediately suggests a statistics associated with it, in making a material point an ensemble of particles must have a certain common characteristic, usually a space characteristic. Consider again a star, to take an obvious example. It is certainly so far away that we can consider it a material point in the sense of Hertz. But this point is now the ensemble of all material particles in the sense of Hertz, having the same space location. The location becomes a statistics like any other, for instance like the energy in the case of an ideal gas. The definition can be easily put in a mathematical form and the logic followed with important consequences. Not among the least ones of these is the interesting conclusion that the light is organically involved in the structure of these material points.

     Speaking of light, it is truly amazing that it offers the most striking examples of material particle and material point. Indeed, Hertz’s definitions allow us logical extensions of these concepts, apparently in contradiction with the intuitive notions they represent. For instance, a material particle is an abstract notion, as noticed before. Therefore Hertz’s definition of material particle is best suited for what Classical Mechanics occasionally called particles of light. Also, a wave surface is the closest picture of the concept of material point. What may be out of order here is the fact that the word ‘point’ intuitively suggests ‘no dimensions’ or ‘zero space extension’. However, Hertz’s definition does not contain the Newtonian factor of extension (the volume), but the idea of “infinitely small space” which is a matter of scale of contemplation of the Nature. So, this definition can accommodate, among the real things subsumed to the concept of material point, extended as well as point objects. And the wave surface or light in general for that matter, can rightfully be placed among such objects.

     This fact is frequently done in Physics, without being explicitly recognized. Take for instance the homogeneity of the matter in the Universe and its space isotropy. One of the highest achievements of the Cosmology in the previous century was in realizing that the two properties are of limited application, depending on the scale of contemplation of the Universe. Namely, the Universe is both homogeneous and isotropic only when we consider it made of galaxies. In other words, only the distribution of the galaxies in Universe is characterized by a constant density. Therefore, at such a scale of contemplation, the Universe appears as a material point whose material particles are galaxies. Still, in other words, any part of the Universe can be considered as a material point having as parts other material points - the galaxies. Now, realizing, in the light of Hertz’s definitions what this means, try to compare it with the fact that the current kinetics of the galaxies in Astrophysics takes these as material particles in the classical sense, i.e. devoid of any structure. It is a poor representation by all standards!

     While we are on this exemplification spree for the concepts defined by Hertz, it’s impossible to forget mentioning the most important example of them all, the one which inspired the kinetic theory in Astrophysics and also the theory of light in its instance as a quantum theory. We are referring to the classical theory of the ideal gas. The model of an ideal gas is the most striking example of the material point in the sense of Hertz: all its constituent particles are indestructible being classical particles. It also contains a striking example of transition from a material particle to a material point, which may be a good source of inspiration for Science.

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