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A Point of Caution: Relation Between Motion and Force

There is a subtle point here that unfortunately has not been exploited along the time because of the prevailing concept of vector attached to force. In order to reveal it, let’s notice that Hertz’s definitions implicitly show that there is a real difference between motion and displacement. This difference enters the definition of material point: forces may act on a material point only through its constituent material particles. These material particles can only be displaced by forces. However the motion may not be a direct consequence of the force as in the Newtonian axiomatics. As a matter of fact strange situations may appear where the point of application of a force acting on a material point is outside of anyone of the material particles from the constitution of that material point. As long as we maintain the geometrical image of vector for a force, like Hertz did, we may not have too much of a choice in overcoming this difficulty but to define further notions which are “concealed” (Hertz, 2003, pp. 223 - 225). We believe that the real lesson to be learned here is that we have to speak generally of a material point in the sense of Hertz when describing a motion we happen to observe, and of material particles in the sense of Hertz when in need to properly describing the action of forces that might go along with this motion. However, when it comes to describing the force as an effect of motion, we need to pay close attention, because some statistics may come into play, as dictated by the scale at which we contemplate the things. After all, a material point is, first and foremost, and ensemble of material particles! And when it comes to ensemble, the statistics is most appropriate method to use.

     When talking of the particle in general within Classical Mechanics one usually understands what Hertz defines as material particle: something indestructible and indivisible. This is why the Classical Mechanics cannot describe the decay of the particles, and this fact came to be considered as one of its deficiencies. This is also the reason why, in a Kepler motion, for instance, it is hard to accept that the action at distance is different from the force that accomplishes it. And yet, this is the case: the action of a planet on the Sun is surely not the same as the action of Sun on the planet. The two actions are indeed equal only as a first approximation so to speak, only insofar as the two can be considered material particles. This far goes the Classical Mechanics. However, the light outside these material points for instance, or their internal temperature for that matter, are the reflection of a certain force inside, acting at the level of ensembles of material particles from their very structure. And these forces give the heat. In the case of Sun the heat is enough to maintain the light, while in the case of Earth it is enough to maintain the life. The action at distance between Sun and planets also includes the mechanical fact that these material points are endowed with proper rotations. The fact that these rotations are an outcome of their space extension is nowadays only a second-hand consequence of the centrality of classical forces.

     The Newtonian view of the Universe, led to the idea that a particle - and from now on we take the term “particle” in the precise sense defined by Hertz - is under the action of the whole Universe. So, when it comes to an actual free particle, then according to the very same Newtonian view of the World, one can see that far from being isolated in the Universe, as usually requested by the First Principle of Dynamics, it is on the contrary under the immanent influence of the whole Universe. This is in fact the natural state of freedom of a particle as pointed out above. It gives strong reasons to the Mach Principle: if the particle is in free motion - and we need to define this freedom by the characteristics of the motion itself - then it is not the absence of forces defining this freedom, but rather their equilibrium in the material particle in question. This is the property that allows us to write the equation of motion of a free particle in the way we usually do: the forces are not absent, they are acting but their resultant is zero. The existence of a free particle can therefore be expressed by the fact that we have, in the point occupied by the particle, equilibrium of forces in any direction in space. Contrary to the apparent limitation, we have here the most direct definition of the action at distance.

     It is then quite understandable why Hertz felt necessary to stress that a particle must be considered as “invariable and indestructible”. However, this is primarily a condition of isotropy of space: the particle is equally affected from all directions of space. The condition can even be expressed more geometrico by the statement that a free particle feels continually two equal forces along any direction in space and these forces are equal and opposite. This way the Third Principle of Classical Mechanics enters the very fundamental concept of this discipline without any assumption regarding the manner in which the space acts on a particle and the metaphysical consequences thereof.

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