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II. Invariantive Mechanics a Short Characterization

The Invariantive Mechanics [1] is exclusively based upon the idea of differential forms. Inspired by the wave-particle duality in the very initial acception of Louis de Broglie [8], it upholds by its postulates what we think is one of the basic characteristics of the free particle dynamics, namely the proportionality between momentum and velocity vectors. This property can be mathematically represented as the proportionality of two differential 1-forms:

(1)

By a theorem of Cartan’s, the proportionality of these two forms is equivalent to the vanishing of their exterior product [3]:

(2)

Here the index k runs from 1 to 4, and we put x₄ = ict, so that p_k and v_k have their forms from special relativity. In fact, starting from the differential 2-form (2) above, the mentioned proportionality is assured only by vanishing of the (4α) components of the matrix of that form (Greek indices run from 1 to 3), if we define, as is usually done in the Invariantive Mechanics, mass as the ratio p₄/v₄. Then there results

(3)

and one can see directly that the classical proportionality of momentum and velocity can be discussed along the same line of argument, upon which we will elaborate later on: the existence of two 1-forms which, in order to be proportional, must have their exterior product zero. This purely mathematical result induces one into the idea that the lack of proportionality between momentum and velocity, which from a mechanical point of view means that the particle is not free anymore, can be taken into account by the existence of a fundamental exterior 2-form. We analyze the consequences of this hypothesis, first for the case of Space-Time.

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