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The Classical Way Around

The Mathematics found here a tool which allowed Physics to consider the problem not in the form that the forces determine the density of matter, but that the density of matter determines the forces: it is the Dirac singular function. The essential property of this function, usually denoted by δ, is given by the following integral equation

Consequently this function can be seen as a distribution concentrated in the point r₀, which shows that it can be imagined from functional point of view as a function having the value infinity in r₀ and the value zero in any other point.

Notice that, if we accept the Newtonian density as an attribute of matter, the Poisson equation defines the force in the same points as the matter. Therefore the Poisson equation characterizes indeed the force as an attribute of a continuum. This fact is in contradiction with the manner in which the gravitational force was conceived for the first time by Newton: a force concentrated in a point. The embarrassing moment has been overcome by using the distributions as above: a mass concentrated in a point is simply characterized by a density which reflects this very fact

Here δ is the Dirac concentrated distribution, this time defined for the three dimensional case: zero everywhere in space, except the point . This mathematical artifact induced the custom of treating the problems of continua as discrete problems without any reserves. For instance the equation (8) can be read as such only in the interior of the mass M, provided the force is universal, i.e. it acts between any two material points, even between the points of a continuum. However here another mathematical artifact, extending the previous one, proved vital: the solution of the equation (12) can be written in the form

This can be easily verified if we take into consideration the identity

The equation (19) was destined to entertain the idea of action at distance in the form of forces acting in space points different from those creating these forces; for our case outside the mass M. This equation makes the Newtonian density of matter an essential ingredient, and certifies once more the impression that the matter is the one that, indeed, creates the force. This impression is in concordance with observation that the force cannot exist without matter. Nevertheless, the equation naturally extends this observation in order to include its dual: there is no matter without force.

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