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II. Considering Some Hidden Forces

There are categories of important phenomena that we cannot judge but by a single component of the motion – usually the radial component – being also unable to associate the other components of motion with one and the same force. This is the case of astrophysical observations, mostly those regarding objects outside the Solar System. For these objects we cannot infer anything about the radial motion but still accept the kinematical description within the classical framework as given by equation (7). In this respect it is to be noticed that vanishing of the right hand side of equation (8) is instrumental in order to obtain a free particle type of radial motion as characterized by equation (9). However, the right hand side of equation (8) can be zero without forces being zero. If such forces exist, they will go unnoticed by the radial motion of the particle, because it still remains characterized by equation (9). However they will be noticeable some other way, for they are non-central forces. Let us see what we can say on this issue. The forces in question depend on the speed and are of the form

where R is an arbitrary function of the position and speed of the particle. Thus, a particle under action of these forces is in a free motion along the line of its position, but its overall motion is not free. In a word, the particle is only partially free. We will illustrate a special choice of the function R(r, q, f), made such that the last two equations from (7) are easiest to integrate. The case we consider is that for which

where R₀ is a constant with the dimensions of a squared velocity. For this case the motion along meridian is that characteristic to a central force:

As to the motion along parallel, we have

Here k and h are two constants of integration having physical dimensions of area rates. The equation (19) can be integrated right away, with the help of (11), to give

under the following convenient identification:

Having this interpretation of the azimuth angle we can proceed to the integration of equation (20). Using equation (21) itself, the result is a matter of direct calculation:

where the integration constant is arbitrary, depending however on the initial conditions on the two angles, and we used the equation (22) for h. It is now directly clear, by a comparison between equations (15) and (23), that this last equation does not represent geodesics on the unit sphere. In a plane parallel to the equator plane, choosing the constant in equation (23) as ±1 as the case may be, we have the following equation of motion:

The appearance of this motion is that of a logarithmic spiral best represented in polar coordinates (Figure 1, at the end of text). The two branches of the spiral correspond to positive and negative coefficients respectively in the exponent of equation (24). Notice the existence of a minimum radius for the finishing points of the motion (the motion starts at infinity). One problem will then be to describe what is going on under the minimal radius of this motion. This kind of motion may be a good candidate for kinematical model of spiral nebulae in case we observe that something really moves along the spiral arms.

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