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I. Introduction

The 19^th century Theoretical Mechanics worked with forces modeling the interaction between particles which have three general characteristics. These are imposed both by the extension of the meaning of observational facts and by calculation purposes. These characteristics are:

1. the forces are central, i.e. they have only component along the line joining the particles;

2. the forces have magnitude depending only on the distance between particles;

3. the forces are conservative, i. e. they can be written as the gradient of a potential function.

Only the first of these properties comes as an extension of an observational fact. This fact is that we are able to notice the action of a force by its consequence, which is usually the change of a motion: the direction of this change in motion is always the direction of action of force. The other two properties are mainly calculation properties, allowing extension of the forces characterizing isolated particles to forces characterizing systems of particles. However, when this extension is made, one usually forgets that the force gets naturally out of the realm of our experience. Here it should have the genuine properties of the action at distance it was supposed to represent in the first place. And the three properties above might not be in this position. In order to reflect the nature of interaction whose representatives are our forces, the three properties must satisfy some extra conditions. One of these comes naturally in our mind in the following way.

     The fact that the extension of the three properties of forces to the action at distance in general reflects the very nature of that action at distance was particularly reinforced in the past times by the fact that the three properties seem to have a “circular property” so to speak. More precisely, if two of them are true, the third is automatically true. This may be taken as the property we need in order to prove that pure speculative properties are natural properties and vice versa. And, for the extension from forces to the actions at distance, this fact is, of course, an extra bonus. However, we know positively that there are changes in the motions, which we are not able to associate with any observed forces. Yet, with the help of calculations, we can imagine actions at distance that might be responsible for such changes. This raises the question if the three properties of forces are indeed equivalent in the way they are supposed to be, in order to be safely transferred as properties of the actions at distance.

     Let us consider a reference frame with the origin located at a certain particle, and any other particle located at the point with respect to this reference frame, these being the spherical polar coordinates. Now, if the force acting between the two particles is central and its magnitude depends only on the distance between particles then that force is also conservative. Indeed, in that case, the force can be written as

where is the unit vector of . It is quite clear that this force is conservative, because it derives from the potential

Suppose now that the force is conservative and central. Then the magnitude of this force does not depend but only on the distance between particles. Indeed, if the force is conservative we have

where V is a potential function. The components of this force are

Because the force is central, we have from the last two equations (4)

which means that the potential does not depend but on the distance r between particles. Consequently, by the first equation from (4) the magnitude of the force does not depend but only on that distance.

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