109

On complex functions analyticity

In Fig. 7 we show the body trajectories in the same field, only the body mass is four times less,  m = 0,25 kg. The same as above, the frequency range can be schematically divided into four bands.

Within that first of a quasi-stationary field, the body moves along the tangent to the line of force at the initial time moment. If the field varied in time slowly ( = 0,2 s^-1), the trajectory displaces clockwise and bends a little, due to small deviation by the lines of force.

Within the second band, the body moves first as if in a channel formed by the field lines of force ( = 0,4 s^-1); then along the equipotential lines with an abrupt repulsion to the field periphery ( = 0,5 s^-1); then it changes the direction of passing by the field core ( = 0,6 s^-1), and finally it passes by the core from the right along the equipotential lines, abruptly repulsing to the weak-field domain ( = 0,65 s^-1). Thus, with the smaller mass but other frequencies of the field variation we trace the same typical trajectories.

Within the third band ( = 0,7 - 2,0 s^-1) the chaotic motion effects on the trajectory much more. At = 2,0 s^-1 we can trace an abrupt turn and the motion towards a stronger field. However the main regularities retain here. The initial sections of the trajectory displace clockwise with the frequency growing.

And at the fourth band ( > 2,0 s^-1) the trajectory displacement direction reverses, and the body transversal oscillations gradually smooth. The natural feature of the motion at this band and with the diminishing body mass is that the body repulsions emerge, when leaving to the weak field; we can trace it by both shown trajectories of this band.

Note that even if we take the mass very small, the above features of motion in a crater-like field will retain as a whole, though the repulsions by the lines of force will impact in this case. To illustrate, we show in Fig. 8 the trajectory of a body having a mass m = 0,0025 kg, 100 times lighter than that previous. Its motion in the fourth band ( 20,0 s^-1) is the most chaotic. However, despite the complicated pattern of trajectories being much alike the Brownian motion, their shape also has some order and regularity. There are clearly seen all four bands described above. As to chaotic pattern of trajectories shown in Fig. 8, it would be interesting to correlate their shape with the prototype - for example, for the case = 40,0 s^-1.

We show the parametrical form of time regularity of the x- and y-co-ordinates of the prototype in Fig. 9, and the trajectory prototype in the plane Z is shown in Fig. 10. In Fig. 9 and especially in Fig. 10 we see quite regular motion of the studied body prototype.

The trajectory prototype in Fig. 10 is a terraced ascending curve, some distorted at the lower level. Notable that in the end of the first level the trajectory prototype passes downwards, outside the belt of the initial one-sheet mapping. The second level is located between the initial belt and that located above, and the third level is located completely in the above belt. Thus we see that the complicated trajectory of the body motion in the plane  W at = 40,0 s^-1  is an analytical three-sheet continuation of the trajectory prototype in the plane Z. This enables us saying of possibility of analytical continuation of many-sheet prototypes when using the dynamical non-conformal mappings.