THE THIRD PRINCIPLE OF DYNAMICS
Author: Nicolae Mazilu
Published on Friday, March 21st, 2008 in category ProtoQuant
Handling the Mathematics: Cartan’s Lemmas
Regarding the practical use of differential forms the interested Reader is referred to the short but comprehensive work of Betounes (Betounes, 1983). It introduces quickly the subject through the most interesting results of Physics. So, the attraction is represented by a 1-form of work of force
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(1) |
where 
is location of the particle characterized by attraction. It is zero in cases where the force is zero or in cases where it is perpendicular to the displacement vector. On the other hand, the repulsion is represented by a 2-form
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(2) |
where “/\” denotes the exterior multiplication of differential forms. For the classical free particle the two differential forms are simultaneously zero: this is the very definition of freedom of particle. Now suppose that we wish to check if a particle is free or not. We have at our disposal the two gadgets above, with which the particle is endowed, and we need to connect them in such a way which allows us to say that if the particle doesn’t have attraction and is free doesn’t have repulsion also and reciprocally. For this we don’t need to go very far with the theory, because the two exterior forms are naturally related in such conditions: there is a theorem of Cartan’s showing that the 2-form of repulsion must be written as an exterior product
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(3) |
where ω is a conveniently chosen 1-form. Only in this case Ω is zero whenever f is zero and vice versa and therefore the definition of freedom is well represented. This gives the proper generalization of the Third Principle of Dynamics: there is no attraction without repulsion and reciprocally. Moreover, the formulation is based on the definition of freedom of particles, showing that this freedom is a function of the environment in which our experience frames the particle, and represented by the 1-form ω. Thus, by this theorem, the 2-form of repulsion is connected to our experience, to our “convenience class” so to speak., as represented by 1-form ω. This last form therefore dictates in turn a certain “repulsion class” giving the current maximal freedom of the material particle.
In this manner the freedom can be defined anywhere, even for a material particle from the composition of a material point. Here however, the freedom is selective so to speak: over the condition (3) we need another condition. In order to formalize this fact for the most general situation, we proceed as follows: assume that the material point whose component is our material particle is analyzed (disintegrated) into its component material points. This means that it is formed from other material points and we need to describe these material points. Then for a certain material point among the component material points the repulsion of particles in the class given by ω is zero, while their attraction still persists. This is a situation where the repulsion is zero without attraction being zero. Consequently for such material particles the left hand side member of equation (3) defining their freedom is zero without anyone of the factors of the right hand side being zero. In this situation we can use another theorem of Cartan showing that this happens if, and only if, the factors of the right hand side are proportional. Thus, for a material point entering as component of another material point, the two 1-forms f and ω are proportional. The meaning of this simple fact is overwhelming. Let’s give a well-known example in order to make it plain.