MATERIAL PARTICLE AND MATERIAL POINT TO HERTZ
Author: Nicolae Mazilu
Published on Friday, March 21st, 2008 in category ProtoQuant
Hertz’s Definitions vs Newton Definition of Mass
Before going any further with our imagination of what has happened, this is the best place to bring Hertz’s ideas into play in his own words. They will literally help us in ‘making the point’. We think the best way to help picture out the situation is to draw directly from Hertz’s Mechanics, as we drew before from Newton’s Principia. These ideas, in the form of a collection of definitions, create the impression that the space characteristics of matter have been defined by Hertz specifically with the concept of action of the force in mind. Indeed, Hertz’s system allows a description of the action of a force entirely characterized by the Third Principle of Dynamics, while leaving free the judgment on the action at distance itself. We reproduce here only the necessary definitions and commentaries (Hertz, 2003, pp. 45 - 46).
“Definition 1. A material particle is a characteristic by which we associate without ambiguity a given point in space at a given time with a given point in space at any other time.
Every material particle is invariable and indestructible. The points in space which are denoted at two different times by the same material particle, coincide when the times coincide. Rightly understood the definition implies this.
Definition 2. The number of material particles in any space, compared with the number of material particles in some chosen space at a fixed time, is called mass contained in the first space.
We may and shall consider the number of material particles in the space chosen for comparison to be infinitely great. The mass of the separate material particles will therefore, by the definition, be infinitely small. The mass in any given space may therefore have any rational or irrational value.
Definition 3. A finite or infinitely small mass, conceived as being contained in an infinitely small space, is called a material point.
A material point therefore consists of any number of material particles connected with each other. This number is always to be infinitely great: this we attain by supposing the material particles to be of a higher order of infinitesimals than those material points which are regarded as being of infinitely small mass. The masses of material points, and especially the masses of infinitely small material points, may therefore bear to one another any rational or irrational ratio.”
Two things are worth noticing, bearing directly on our subject matter here. The first one is the definition of mass. We consider it in order to show what it occasioned and what actually Hertz’s definitions targeted. The definition of the mass to Newton is (Newton, 1995, p. 9):
“Definition I. The Quantity of matter is the measure of the same arising from its density and bulk conjunctly”
Neat and very explicit, isn’t it?! It’s like a vicious cycle: in order to know what quantity of matter (the mass) is, we need to know what density is. On the other hand, in order to know what the density is we need to know what the mass is, because the density itself also “arises from mass and bulk conjunctly”. But, we shouldn’t take things the hard way! Fact is that in the times of Newton the density was much closer to the Natural Philosophy than it became later when the Man started asserting a better understanding of the structure of matter. The density was then an attribute that could be felt, it addressed to senses. We can feel that the water is denser than the air, that a stone is denser than the water, etc., and we can even create an approximate scale of densities based only on our senses. Moreover, the Archimedes’ principle had already been known for almost two millennia, thereby providing a method of measuring the densities, at least for solids. Even today we use the old scale of densities that takes the density of water as unity. Seen in this light, the definition of Newton is perfect, and led to some very important consequences.
One of the most notable consequences of this definition is the fact that it occasioned a definition of the force as a vector field, i.e. as a state of space containing matter. This fact has been fruitfully exploited by Maxwell, for instance, in creating his electromagnetic theory of light. The basic philosophy here, logically derived from geometrical analysis, is that the force in a portion of space is related to the density of matter from that portion by the so-called Poisson equation, which can actually be used to calculate that density. This way the density of matter is directly related to the forces which can be ascribed to it. Of course, this formalism has drawbacks, like any human endeavor. For instance, it shows that the Newtonian force, going inversely with the square of distance can be considered as universal only as long as the density of matter is zero. In fact it is the only force compatible with a zero density of the matter. The way this conclusion came to light can be seen first in the modern Cosmology. It is inferred from astrophysical and astronomical observations that the matter in the Universe has a very small mean density of the matter in the Universe (≈ 1.0×10-27 kg/m3). This was deduced by considerations independent of force, from the region of the Universe accessible to our observations. On the other hand, we need to recall the same space behavior for the forces between the particles representing electricity. This fact was discovered by Coulomb. The electric particles, again, enter the structure of any known continuum, but in an electrically neutral continuum the charge density is exactly zero, not approximately. But the most important predictions of this theory are that the only forces compatible with a constant density are the elastic forces, and a zero density is always unconditionally compatible with noncentral forces. These two consequences of Newtonian theory, if properly combined would lead directly to the theory of light even within the framework of the Classical Mechanics. As it happened though, the theory of light took a wide turn, described here in the previous chapter, specifically with the purpose to avoid the Newtonian Dynamics.
There are a lot of consequences of Newtonian Dynamics; some are good some are not quite so good, but this is a matter of taste, for all of them are truths revealed by Mathematics. One of these consequences is that Science has always considered the density as a primitive quantity, which determines the forces through the Poisson equation, not that it is determined by them. This, in our opinion, is not quite so good, for the density depends on the space scale we consider when describing the Universe around us. It is thus apparent that we have either to avoid the Newtonian definition of mass, which makes mass actually a density, or to avoid the concept of force, or both. Coming back to Hertz, one can say that he took this last option. As one can see from excerpts above, he uses a kind of Cantorian definition involving the powers of continuum closer to, say, the modern theory of probability. He even underlines this fact once more by the observation that the material points are to be supposed of lower order of infinitesimals than the material particles, which thus get the status of the highest order of infinitesimals. Then, in the development of the treatise just cited, he introduces the theory of constraints, by generalizing a previous idea of Gauss, thus assigning the inherent interactions of bodies to legitimate mathematical instruments.