# Why are velocity dependent forces considered non-central?

I was going through Classical Dynamics of Particles and Systems (Thornton-Marion, 5ed). On page 50 it is stated:

Any force that depends on the velocities of the interacting bodies is considered non-central. Velocity dependent forces are characteristic of interactions that propagate with finite velocity.

Now, I do not understand this at all (but I do understand what central forces are). First of all, what does "propagate" mean here? What "interactions"? Can someone provide an intuitive explanation of this statement?
The book further goes on to give an example for moving electrical forces, which I do not understand as well.

Thus the force between moving electrical charges does not obey the Third Law, because the force propagates at the velocity of light.

Also, as a side question, does this mean that drag force is non-central, since it depends on the object's velocity in the fluid?

## 1 Answer

The usual definition of a central force is a force such that, with appropriate choice of coordinates, it can be written $$\mathbf F = f(\mathbf r) \hat r$$ where $$\hat r$$ is the unit vector pointing away from the origin. Some authors make a stronger definition, that $$f(\mathbf r)$$ can be written as $$f(|\mathbf r|)$$, so the force only depends on the distance to the origin.

The idea is simply that this condition is a very large and useful simplification. If we consider forces in general, we can't do very much other than write down Newton's laws and stare at them. If we assume that the force is central, we can start talking about what kinds of forces will produce what kinds of orbits, whether they can be written as the gradient of some potential, etc - all of the things which are subsequently discussed in that chapter. Central forces are very important, both as reasonably good models for real forces like gravity and the electric force within an atom and as a way to gain familiarity and intuition for some of the general tools utilized in mechanics, so it makes reasonable sense to (a) define them this way and (b) devote a chapter to their study.

The example given by your textbook is pretty standard. If I have two electric charges separated by a distance of 3 meters and then I shake one of them, the force on the other doesn't change for $$\frac{3\text{ m}}{c} \approx 10$$ ns, because wiggles in the electric field propagate at the speed of light. For that reason, the forces on the respective charges will be different for that nanosecond, and so the third law is violated. Of course, if you're only interested in slowly moving particles at relatively short distances from one another, this violation can be ignored without incurring too much error.

Also, as a side question, does this mean that drag force is non-central, since it depends on the object's velocity in the fluid?

Well yes, but I'd say the more important issue there is that the drag force in a fluid is not always directed toward some central point.