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There is an important consequence of the Corollary 3 of the Proposition vii of Newton (Newton, 1995; p. 48) that may help changing our usual image of the electromagnetic interactions and particularly of the propagation of light. In obtaining the formulas giving the Newtonian central forces, everything is referred to the case when one of the two centers of force is the center of the ellipse representing the natural trajectory, for in that case the force is known

to be the elastic force. The usual Newtonian force is obtained when the second center of force is in one of the foci. But, in general, even if we are dealing with central forces, their magnitude is not exclusively dependent only on the distance, it also depends on direction, and therefore the second center of force can be anywhere in the plane. If, therefore, in the usual geometric picture of that corollary, the center of force is considered outside the ellipse, in O say, the arguments are still in force. By Hamilton theorem the point P acted upon by a force directed towards point O describes an ellipse naturally delimited in plane by the two tangents OT₁ and OT₂ from point O. Indeed, assume that the equation of the ellipse is of the form

(1)

The geometrical interpretation of this equation (Salmon, 1904) shows a conic, tangent to the two straight lines

(2)

both passing through the pole of the straight line

(3)

with respect to the conic section. Using equation (1) in the analytical form of Hamilton theorem, the Newtonian force ends up showing like

(4)

Under the action of this force a material point describes, indeed a conic tangent to the two lines from equation (2). This can be shown, in a special case, by the second principle of Newton, thus making this principle compatible with the practical definition of force. It will be an interesting exercise to repeat here the proof of this assertion, just for the sake of argument, even though it was done a few times in the history of science (Appell, 1893; Routh, 1898). Unfortunately it may remain only a historical curiosity, for today Newton is only judged by real or, quite so often lately, imaginary ideas he might have had regarding the space and the time. It is altogether forgotten that he extracted these ideas from the necessity of defining the force which, to him and a few others, was more than the mere cause of motion or its effect.

Assume therefore that the second principle of dynamics gives us the liberty of writing the equations of motion in the form

(5)

where μ is a constant, and a dot over symbol means derivative upon time. It is then obvious that the coordinates defined by

(6)

have formally the same equations of motion, i.e. the second principle is invariant to a linear transformation of coordinates of the moving material point. Specifically

(7)

Here it is obvious that if c₁ and c₂ are not zero, the acceleration is not central. Therefore the second principle is not valid in the coordinates ξ, η. Thus, even though from a geometrical point of view it makes no difference whether we choose those two constants as zero, from physical point of view is instrumental that they should be zero. This is not just an isolated problem. It has an essential imprint upon the whole classical Newtonian dynamics recurring, at times, in relation to space-time description of Nature: it is always necessary to make the description in a privileged reference frame. It was one of the main reasons of building the relativity, both special and general. Let’s therefore assume that c₁=c₂=0.

In that case, following the classical routine, we use polar coordinates

(8)

in order to obtain the Binet’s equation

(9)

Here the accent denotes derivative on angle θ. The general solution of this equation is

(10)

where m₁ and m₂ are two constants to be given by initial conditions of the problem of motion. Reestablishing here, first the coordinates ξ, η and then the coordinates x, y, gives the form (1) of the trajectory of motion, of course, with the center of force situated in the origin of the reference frame.

This occasion asks for unfolding of some forgotten history (Roche, 2001). The action of magnetic field is characterized not by a direction, but by a plane of action – Ampère’s principal plane – in which plane one can perceive that action by a rotation of the currents. If, for instance, CP in our figure is a dipole, the trajectory of P reflects the action of a magnetic field, inasmuch as it is manifested by a rotation in plane of the position of P with respect to C. From physical point of view the magnetic field is manifested as usual: central forces towards O (the regular magnetic attraction or repulsion) together with a rotation in the plane of action.

This physical image is precisely the one emerging from Maxwellian space time electrodynamics. For instance the case where O and C are very far away, i.e. the angle between the two tangents OP₁ and OP₂ is very small, gives a light ray in the manner of Fresnel. But, if it was ontologically possible, we have here the means to find, on one hand, the roots of this possibility in the very definition of Newtonian forces and, on the other hand, the roots of the possibility of the electromagnetic image of light.

REFERENCES
Appell, P. (1893): Traité de Mécanique Rationnelle, Gauthier-Villars et Fils, Paris, Tome I
Newton, I. (1995): The Principia, Prometheus Books, Amherst, New York

Roche, J. (2001): Axial Vectors, Skew-Symmetric Tensors and the Nature of the Magnetic Field, European Journal of Physics, Vol. 22, pp. 193 – 203

Routh, E. J. (1898): A Treatise on Dynamics of a Particle, Cambridge University Press, Cambridge, UK

Salmon, G. (1904): A Treatise on Conic Sections, Longmans, Green & Co., London