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Motion of a Charged Particle

Indeed, the material points also possess electric charge, not only mass, and in this case is very easy to see that the proportionality in equation (8) is occasionally destroyed, in a magnetic field for instance. Let us formally follow this process. The 1-form (7) for attraction is not satisfactory any more, and a first reaction would be to redefine it still retaining, however, the inertia through the mass of the material point. Thus instead of (7) we will redefine the 1-form of momentum by

In case this form is zero, i.e. there is no attraction, we still have the definition (8) as a natural case. However, let’s assume the inverse case, namely that there is attraction expressed by this 1-form. In this case, we have also repulsion, whose differential 2-form is defined by

This form allows us to still define the freedom of the particles by the equation

where

is a conveniently chosen differential form. Thus the discussion translates here from the “velocity class” in the “class of vector “. In a collision of such material points, the free material particle is defined through equation (11). One of the component material points is defined by the absence of repulsion in this class, which leads to the proportionality of the corresponding vectors:

Now it becomes clear what the vector is, because this expression of the momentum is classic in Electrodynamics: we have here the vector potential of Electrodynamics. Moreover, the repulsion 2-form has now identity in our experience: it is the differential form generated by the induction of the magnetic field, whose coefficients can be written as

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