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Well-Known Example: Second Principle of Classical Dynamics

Let’s “conveniently chose” the 1-form ω as a form built upon the acceleration vector of a material particle:

so that for the components of repulsion from equation (3) we have

These coefficients are the components of a vector product which, as well known, is geometrically a vector perpendicular on the plane of its two factor vectors. This is the essential condition of the Newtonian free particle: it is characterized by a vector always oriented perpendicularly on the plane formed by the acceleration vector and the attraction force vector. Pretty strange according to our intuition, which educates us that the repulsion is a force and that a repulsed particle must move to infinity, but this example is among those which can show us how to educate our intuition based on Science. This vector product is the expression of the existence of repulsion for the particle involved in Newtonian inference of the gravitational force; this is how the particle can be identified in the “class of acceleration” so to speak.

Let’s assume now that the freedom is redefined: among the situations where the conveniently chosen form can be constructed on the acceleration vector (”class of acceleration”), there are some for which the repulsion is zero without the attraction being zero. In other words, the particle undergoes attraction without however being able to repel all the other particles. In this case the repulsion cannot be zero but only in case the two vectors, factors of the vector product are collinear, as equation (5) implies. This is the real meaning of the proportionality of the forms f and ω in the general definition (3) of repulsion. Thus, if the repulsion is zero in the space realm of a material point where the attraction is not zero, then the two vectors, force and acceleration must be proportional, i.e. we must have

It is easy to recognize in this equation the Second Principle of Newtonian Dynamics. Here however, this principle is more natural: it relates the acceleration to the attraction force, not to a force in general as Newton postulated. The force in itself is not a concept, but only a fact of experience, intuition in the sense of Kant. It is given to us through our tactile sense and is measured, at its origin as a concept, by the attitude of response of our body.

Let’s now philosophize in the good old Newtonian manner, with the help of this mathematical basis. Consider the solar system: it is a material point in the sense of Hertz. Its particles were historically defined through acceleration by Newton. In other words, any particle we might be able to imagine must satisfy the equation (5) according to our experience. The material points, components of the solar system are the Sun, the planets, and all the other objects forming its material, whose number is continuously increasing day by day, due both to technology and to the increasing number of amateur astronomers involving themselves in watching the sky. Consider now one of these material points, a planet, in order to be in line with the history of the subject. The particles of such a material point have a limited freedom, which needs to be redefined in the “class of acceleration”. Because this material point is a whole, between its material particles there is repulsion in the “class of acceleration”. Therefore the equation (6) takes place where m is the mass of material particles, and this relation gives us an operational manner of defining the force, fully exploited by Newton when he characterized the force of gravity based on the centrifugal acceleration.

The history proves however that the “class of acceleration” is not sufficient for describing the solar system. First of all, it has been seen that, in the description of the motion of planets in this way, the center of attraction does not coincide with the center of motion, and this means that the force might not be quite proportional with the acceleration. In order to save the Second Principle of Mechanics, a struggle has been started even by Newton, but with results undecided even today. Fact is that the principle cannot be saved! Indeed the vectors of acceleration and force are not collinear, and this is quite obvious both by the fact that the planets’ motion is not in general plane motion and by the fact that the material points representing them have rotation motions with a component perpendicular to the plane generated by the vectors of attraction force and centrifugal acceleration. Before entering in some details here, let’s give an example of another “class” which served and still serves the Science today, and thus confirms this general Principle of Attraction and Repulsion. This time, however, the attraction and repulsion are not realized by forces and this will be explicitly seen.

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