# 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?

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 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.