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III. An Obvious Place to Find Noncentral Forces

Previous argument may open doubts as to the existence of noncentral forces. As they have never been used in explaining at least one observed motion, we have reasons to believe that this is actually a consequence of the limited mechanistic point of view usually adopted when carrying discussions about observed motions. Indeed, this point of view places from the outset the source of forces in a well defined particle and, as always in this kind of problems, it is hard to see just how the forces are created without perceptible material constraints. As a matter of fact it is very hard to see how all the forces acting at a distance are created without any constraint. This is, for instance, why it took so long to Newton to realize the identity between the daily weight and the force holding the Moon on its trajectory. Our subject forces are so much harder to comprehend, as they have this extra oddity of not even acting like the weight, i.e. along the line connecting bodies, in order to allow an obvious inference for their existence. However the things change if we adopt a larger perspective for the judgment of these forces.

     First of all, there is nothing to add to the above discussion, in order to consider such forces as a realistic model for the planetary system. We underline planetary system because decidedly it is not a Kepler model as described by the Newton’s equations of motion. First of all our forces here are not central forces. They have in common with the central forces case only the fact that they are conservative. Secondly, our forces have a component along the line joining the particles, namely

As usual, this indicates that the system is not a stable one. The potential V_r(r) from (12) is then a measure of the mechanical work dissipated or absorbed when the two particles move along the line joining them. If this work is not zero, then the system either collapses or disintegrates. Here, however, we get the possibility of describing a stable system: that system corresponding to V_r(r) = 0 which, by (24) gives the magnitude of force as

This kind of forces do not have but components orthogonal to the line joining the two particles: the two particles neither attract nor repel each other, but remain stationary at the same distance from each other.

     It is interesting to see what the difference between this system and the Newtonian one is. Remember that the Newtonian system is the dynamics which demonstrates the Kepler laws. For comparison we take a particular case compatible with the Newtonian one, namely the case of the equator, θ = π/2. Let’s calculate the components of acceleration in the polar coordinates of that plane for the particularly instructive case where the force on meridian is zero. This condition assures the planeness of the trajectory. We have

where the dot means differentiation on time. Here we also used the Newton’s second law to the effect that the acceleration is proportional to the applied force. The last equation (26) can be integrated immediately because it has r as an integrating factor. Multiplying through by r, we have

The quantity , representing the area swept by the radius vector in a unit of time, is a constant for Kepler motion, the so-called area constant. Actually this is the case of any motion under any central forces. Then the equation (27) shows that this rate is no more a constant in the present case. The constant γ gives the rate of its variation.

     If we use the Newtonian equations of motion, this is the essential difference between this type of forces and the central forces responsible for the Kepler motion. The equation for r can be obtained directly from the first equation (26) and (27). It is

where t denotes the time parameter. This equation is of the type usually known as Emden-Fowler equations, and it does not seem to have a solution in a known closed form (Leach et al., 1992). Actually, we cannot expect a closed form solution because the trajectory cannot be specified the same way as in the case of central force system. This fact copes with the known Bertrand theorem (Whittaker, 1988) showing that such a force has no closed trajectories, even in the case where it is a central force.

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