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II. An Ignored Property of the Action at Distance

The last case, when the force is conservative and depends only on the distance between particles is the most interesting one (Burns, 1966). These two properties are purely speculative properties, and are basically used for calculation purposes, one from energetical point of view and the other from dynamical point of view. Therefore they are used in any extension of the individual forces to ensembles of particles characterized by those forces. And if they imply logically the first property, which is an observational property, then they are indeed properties of the actions at distance which the forces are supposed to represent.

     Unfortunately this is not the case: in calculations something always remains hidden because these forces are not necessarily central. In other words, the direction of force does not necessarily coincide with that of the action at distance it represents. This can be seen from the fact that the equation

(6)

embodying the above two properties of conservativeness and distance dependence, has nontrivial solutions for which all three components of force as given by equation (4) are non-null. We cannot give the most general solution of (6) but only prove our assertion by providing a particular example indicating the existence of a broad ensemble of such solutions. Namely, by equation (4) the equation (6) can be written as

(7)

Now, we solve this equation by writing the potential as a sum of three terms, each one of them depending on only one coordinate, as in the usual method of separation of variables for the classical Hamilton – Jacobi equation (Whittaker, 1988). We have

(8)

Inserting (8) into (7) we obtain the differential equation

(9)

where the prime denotes differentiation with respect to the argument of function. This equation can be separated, first in the form

(10)

where β is a real separation constant. Then we get the following two differential equations

(11)

The first one of these equations gives the part of potential, depending only on r, as

(12)

while the second one can further be separated to give

(13)

where γ is a new real separation constant. This equation provides the other two parts of the potential function, namely

(14)

Using (12) and (14) in (4) we find the components of force as

(15)

On these equations we can read the following important conclusions. In principle, there is a central component of force between the two particles, as given by (15₁), but there are also components of force along the meridian and parallel, as defined by (15₂) and (15₃) respectively. Now, it is quite clear that these components of force are not real for the whole range of the coordinates r and θ, if the separation constants are to be always real. For reality, the following conditions must be satisfied:

(16)

The second of these conditions imposes restrictions on the range of variation of the angle θ, namely

(17)

This tells us, first of all that for the equality sign in (17) the component of the force along the meridian is zero. This means, secondly, that the force has non-null component along the meridian only for a belt about the equator, defined by

(18)

where θ₀ is the solution of the trigonometric equation

(19)

Then, the existence of a component of force along the parallel, namely F_Φ as given by (15₃), allows the following image of the relative motion of the system of two particles: suppose, for definiteness, that we take the particle from the origin of the reference frame as fixed. Then the second particle is pushed to move around the origin by a force having the magnitude

(20)

This same particle also has an “oscillatory” motion, imposed by the force along the meridian having the magnitude

(21)

This force has a maximum

(22)

on the equator while on the limits of the belt (18) it has a minimum. Thus, the second particle, moving around the central one, is pulled up and down with respect to the equator. When that particle is on the equator, the total non-central force has the value

(23)

Now, this description is, of course, an ideal case in that the reference frame is vague to say the least. For instance, we could choose the origin of this frame in the other particle as well. The description then would be the same.

     There is no way to distinguish between the two particles. If we are able to endow the particles with masses, then we can consider the origin of the reference frame as located in the common center of mass. The description of the motion of the two particles around the center of mass is, qualitatively, the same as above. But choosing the origin does not mean that we removed the indeterminacy of the reference frame: it still remains indeterminate with respect to its orientation. The only way to remove a part of this last indeterminacy of the reference frame is to determine it by the force itself. Thus, if we are able to measure F_θ and F_Φ for a system of two particles, these components must have the above properties and we can choose the equator plane as a plane where F_θ and F_Φ are extremes (one of them is maximum, the other minimum). Once this plane is detected it gives a reference frame up to an arbitrary rotation about an axis perpendicular to that plane. As mentioned above, the indeterminacy of the reference frame still remains but this time it is a real advantage: the arbitrary rotation in the equator plane accounts for the indeterminacy of the pair of particles situated at the same distance from each other, in a physical system for instance. Indeed, in a real physical system what we just described as a two-particle is actually an “equivalence class” for that model. We must assume that each particle can be described as being a part of a class of two-particles, a class being characterized by a certain fixed distance r between its two component particles.

     This property of the actions described as forces, of having a possible noncentral character has been largely ignored in the specialty literature. The primary consequence of ignoring was the lack of motivation for a deeper analysis of the concept of physical reference frame. The result is that this concept has been tied up with kinematical properties, related to another aspect of corporal matter, namely the inertiality of its motion. Strange enough, a deeper study of noncentral forces also leads directly to the idea of inertia, but from the dynamical point of view initiated by Mach. This is the subject of the next sections of present essay.

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