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A Mechanics With Reference Forces

Imagine now that Newton was inspired, in postulating the Law I, by the light. This fact is entirely plausible, because the light satisfies all of the requirements of this postulate: no sensible force acts upon light and, if one assumes that it is given by particles, then the motion of these particles is uniform and rectilinear. The problem here is that of forces: they don’t exist for light, so they should be inexistent in general in such a system. Therefore when postulating such a particle - which Newton didn’t - one has to eliminate forces from the system - which Newton couldn’t - and this was indeed the declared task of Hertz. Both of them, Newton as well as Hertz, were consistent with their own logic, and actually with the logic in general, as a universal way of philosophizing.

     What then the Mechanics would look like? Well, a little deeper maybe, but just as simple as its classical counterpart; anyway not along the lines of Hertz himself. We can offer here a tip about it. Instead of referring it to the motions we observe, we have to refer it to the forces per se. In a continuum theory the forces are described as fields of vectors. We have to recognize the light as the action at distance without forces, therefore as a situation characterized by zero forces. It is a limit situation, which has to be considered as invariant all through any theory of forces. This means that any problem involving forces must be indifferent with respect to light. It’s just a way to set the problem; if it wouldn’t be indifferent, we would have the light entering the problem of forces in the first place. Now, the Fresnel’s theory revealed a transformation which is vital for the space forms that light could take. We don’t think this transformation appeared just by a geometrical chance: it reflects the very essence of our problem. For, in a physical problem involving forces, like in any physical problem for that matter “any circumstance left unspecified in the statement of a problem defines an invariance property, which the solution must have if there is to be any definite solution at all” (Jaynes, 1973). Indeed, a certain circumstance cannot be left unspecified in the formulation of a problem but only because our ignorance. And this ignorance is due to the fact that the senses cannot tell us anything about that circumstance. If it still passes unnoticed in the correct solution of the problem, this means that the problem, as well as its solution, is invariant with respect to that circumstance, thus reflecting correctly the capabilities of our senses. In the theory of light we know nothing about forces; therefore the theory must be invariant with respect to the situation of zero forces. Now, in terms of continuous fields of forces, the forces of zero magnitude can be expressed by a quadratic differential equation, using the concept of potential. This equation is indeed invariant with respect to the transformation by inversion which gives us the wave surface of Fresnel (Campbell, 1903). A Mechanics satisfying to such invariance can only be based upon Hertz’s concepts regarding the material particles and material points.

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