THE THIRD PRINCIPLE OF DYNAMICS
Author: Nicolae Mazilu
Published on Friday, March 21st, 2008 in category ProtoQuant
The Philosophy Behind a Mathematical Choice
Therefore the existence of repulsion, of “keeping at distance”, is a necessary condition for the maintaining the identity of a particle, involving all the particles of a material point. All of them are kept at distance by everyone of them, and all of them keep this one apart. We are talking of “keeping at distance” where many of us think of repulsive force, but we hesitate to use this term because it suggests the force even in the cases where the repulsion is not manifested by forces. The mathematics that will be suggested here will show clear where we have to direct our thinking. Therefore the repulsion can only be thought of as a flux of forces. The geometrical gadget attached to the particle from this point of view is a differential 2-form (Hehl, Obukhov, 2000). Therefore, when inside a material point a material particle is characterized by a differential 2-form. This form is zero for the isolated material particle: the isolated material particle has no repulsion of any form.
On the other hand, the attraction between particles is directional and selective, in order to make possible the disintegration of the material points. As a matter of fact the directionality is perhaps a too strong determination. It is enough to say that our experience does not show that the attraction needs a surface as its essential condition of existence, because it can be accomplished directionally. In a material point, the attraction can only be thought of as the action of a force along a certain direction, as Newton did, or as simultaneous actions of such forces. The geometrical gadget which couples the force with a direction is the differential 1-form. Therefore, inside a material point a material particle is characterized by a differential 1-form, representing attraction. This form is also zero for the isolated material particle: the isolated material particle has also no attraction of any form.
The free material particle is thereby literally the ground zero for the repulsion as well as for the attraction. As a matter of fact, this feature - the freedom - is the defining characteristic of a particle in any environment. Only, the definition of freedom strongly depends on environment as actually even our daily life experience shows. No matter of what the environment is, the freedom can be defined by a natural connection of the two geometrical gadgets with which the particle is endowed. This connection is given by a theorem of Cartan relating differential 1-forms to differential 2-forms or, to be more correct, vice versa (Creangă, Luchian, 1963). Let’s express this theorem for our current necessities.