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Newtonian theory of forces responsible for natural motions doesn’t necessarily assume that these forces have the magnitude exclusively dependent on distance. The only assumed fact was initially that these forces are central, for this is the only thing in accordance with our immediate experience regarding forces. Therefore, eliminating the unnecessary restriction of the dependence of force exclusively on distance between bodies, might give us the possibility to uncover things which that assumption has hidden from us. Indeed, the Newtonian theory of forces describes, for instance, the double stars. It can also describe the magnetic action. Assumptions of dependence only of distance hid these facts of observation, for such dependence is only related to special points from the plane of the observed orbits. Let’s take this freedom for describing here the interaction of a center of force with the electrons, assuming that this center of motion is moving. The image of instantaneous interaction is given by the geometry of magnetic action upon a charged particle. The rest is geometry.

Assume, indeed, that the equation of the ellipse is of the form

(1)

 It is worthwhile using the Dirac notation for the following calculations, both for its handiness in formal calculations and for suggestive interpretation of the geometrical results. In this notation the position “ket” vector and its corresponding dual “bra” are given by the matrices

(2)

 The distance is then given by the norm of this vector

(3)

A translation of the vector is represented by addition:

(4)

while a general homogeneous transformation comes to a left or right multiplication with a 2×2 matrices:

(5)

Here we will use the bold letters exclusively for matrices.

With these notations the equation (1) can be written as

(6)

with the following identification:

(7)

 The center of this conic section is given by the formal equation

(8)

If we refer the conic to its center by the translation

(9)

 then its equation becomes

(10)

Then the problem of a moving center of force can be simply reformulated by finding the set of conics of the same center. Indeed, we represent the condition of “same center” by dx_c = dy_c = 0, which, in our present notations mean

(11)

 or, after using the known formal relation da^-1 =-a^-1·da·a^-1,

(12)

 Therefore the condition of fixed center comes to an evolution of the vector (a₁₃, a₂₃) managed by the matrix of the quadratic form from the equation of conic, and its differential. In detailed form equation (12) sounds

(13)

The matrix of evolution can be further adjusted to a special form

(14)

where

(15)

 are three differential forms generated by the entries of the matrix of quadratic form from the equation of conic and their differentials.

Now, in order to have an idea of what this evolution means, we need first to have an idea of what its subject mean. The conic (1) is usually referred to the center of force as origin. The polar line of a point of coordinates x, h can be obtained directly from the equation of the conic, by a polarization procedure, and its equation is

(16)

 For ξ = η = 0 the equation of this line is

(17)

 showing that (a₁₃, a₂₃) are the tangential coordinates of the center of force with respect to our conic section. Therefore, the equation (12) represents, even though not quite in a direct way, the evolution of the center of force as we “read” it in the variation of the local conic sections. The problem is: what do we read in that variation?

Well, not too much to start with, but quite enough if we are able to integrate somehow the equations of evolution. First we read that, due to the evolution of the center of force, we have a host of local conic sections of the same center. We may take this statement in the reverse, to mean that the host of local conics is a direct reflection of the evolution of the center of force. Either way, the final decision on that meaning hangs on the possibility of integration of the matrix of evolution (14), i.e. to obtain a matrix like

(18)

 Here Ω_k are some functions obtained from the corresponding differentials ω_k, by the procedure of integration, whatever that procedure might be. Indeed, in that case we can write (Bellman, 1960)

(19)

 The matrix exponentiation can be performed in the usual manner using the formal series of matrices, with the (formal) result

(20)

where we denoted

(21)

Thus, we can have the evolution of the center of force, pending the definition of integration. For this we have a few choices, but the most attractive of these relates to a suggestion found in the geometrical treatment of the initial conditions of the Kepler problem.

Fact is that the set of symmetric matrices related to our quadratic forms can be organized as a Cayleyan space with respect to the Absolute represented by the singular matrices:

(22)

In this image every matrix is represented by a point in space, having the coordinates (a₁₁, a₁₂, a₂₂). Matrices of positive determinant are represented by points inside the Absolute, while those of negative determinant are represented by points outside the Absolute. By the same token, in terms of conic sections, one can say that the points inside the Absolute represent ellipses the points of Absolute represent parabolas and the points outside the Absolute represent hyperbolas.

Now, if we refer the points inside the Absolute to their asymptotic directions, one can have the following parameterization of them

(23)

 where (u ± iv) are the (complex) slopes of those directions, and D any positive number. In this parameterization the differential forms from equation (15) are:

(24)

 Therefore they don’t depend on D. The Cayleyan metric of this space is provided by

(25)

 This reveals the metric of hyperbolic plane, and suggests an interesting question: what would be the sequence of matrices corresponding to the evolution of the parameters u, v along the geodesics of the hyperbolic plane?

It is known (Guggenheimer, 1963) that the geodesics of the hyperbolic plane are half-circles of the upper complex plane, which can be parameterized by

(26)

 Here u₀ and v₀ are the values of u and v at t = 0. The parameter “t” itself turns out to be the arc length of geodesics as given by equation (25). Along those geodesics the differential forms (24) have constant rates

(27)

 so that the integration from equation (18) can be defined in a regular way. If one considers the integration starting from t = 0, then

(28)

 and the exponentiation from equation (20) gives

(29)

Therefore the equation (19) can be written in the form

(30)

 showing that the tangential coordinates of the center of force are on the following locus

(31)

which is obviously a hyperbola.

The results thus far seem to point out the following conclusions: if the center of a force is in motion, and this motion is a hyperbola, then there is local family of ellipses determined by this center of forces in its different positions. Reciprocally, in a local family of ellipses one can read an outside disturbance due to a center of force moving along a hyperbola. This is, for instance, the case of scattering of alpha particles that led to the Bohr model of atom, and to the subsequent quantum mechanics. But here the theory is purely geometric, with no reference to a nucleus: the geometrical shape of the motion of electrons is referred only to the moving alpha particle, as being the sole center of force.

REFERENCES

Bellman, R. (1960): Introduction to Matrix Analysis, McGraw-Hill Book Company, Inc., New York

Guggenheimer, H. W. (1963): Differential Geometry, McGraw-Hill Book Company, New York