V.4 No 1

43

On orbital stability of oscillators

To facilitate the visualisation of field of revolving proton, we will use some other method to plot, paying attention that proton, so to say, radiates at each moment of time only one line of force. In this case the following theorem will be true.

Theorem 1. If the proton orbited with the velocity v_p < c , at each moment of time and at any arbitrarily chosen point N of space surrounding the field source, there is present the excitation from only one location of proton in its orbit.

To prove it, choose in the orbit two arbitrary locations of nucleus, as it is shown in Fig. 3.

Fig. 3. Sketch to determine the possibility of excitations coming simultaneously from several locations of proton in the orbit

Let the time, which proton needs to pass from its position A to B, be equal to t_AB,v 0 . In order at the point N to be present simultaneously two excitations from points A and B, it is necessary that

(19)

where t_AN =  r_AN /c is the time interval during which the excitation from A reaches N.

Now join A and B by the chord. It forms   ABN . According to basic theorem of geometry, "any side of triangle is less than the sum and more than difference of two other sides" [7, p. 269]. So

(20)

If we divide the right and left parts of (20) into the velocity c of excitation propagation in space, we will yield

(21)

where

From the sketch in Fig. 3 we see and from geometry know, the length of arc AdB along which the proton moves is more than chord AB, so we can write down a chain of inequalities:

(22)

Given (21), this brings us

(23)

It proves the initial condition (19) unrealisable, if the points of field radiation are different when it moved along the orbit. This proves the theorem.

With this theorem we can plot the diagram of field of scalar potential of revolving proton as the equipotential rings excited by this proton when it orbited. The size of these rings will grow with the distance from the proton location at the current moment of time t₀  to the point of radiation of corresponding equipotential line, as it is shown in Fig. 4.

Fig. 4. The principle of plotting of equipotential lines of dynamic scalar potential excited by the revolving proton

We can easily extend such technique of field visualisation to any region of our interest around the proton, if we add the equipotential lines, corresponding to the neighbouring periods of studied dynamic process. The size of studied region will be proportional to the number of wavelengths propagating from the proton into the surrounding space.