How to get statics out of a dynamic force concept? If one defines force as the time derivative of momentum, i.e. by
$$
 \vec{F}= \frac{d}{dt} \vec{p}
$$
how can this include static forces? Is there a generally accepted way to argue in detail how to get from this to the static case? If not, what different solutions are discussed?
Edit: 
I shoud add, that by static forces I mean forces involved in problems where bodies don't move.
 A: I don't exactly know what you mean by static forces. But I am going to take a wild guess here and assume that by that you mean forces involved in problems where bodies don't move. I think you assumed that Newton's second law quantifies a force. This is actually wrong. First of all realize that a force is an interaction and it still acts whether the body on which it acts moves or not. Newton's second law quantifies the total effect of all such forces on a body of mass $m$ and not the force itself. For example the Newton's law of Gravitation tells you that the force between two masses is:
$$\vec{F} = G\frac{Mm}{r^2}$$
Now this is practically useless unless you specifies what a force does on a body. That's where Newton's second law comes in. So along with Newton's second law, you have a complete theory of (classical) gravitation.
Also the $\vec{F}$ in Newton's second law is the total force acting on a body having momentum $\vec{p}$. So when bodies don't move the net forces on them is zero. But that does not mean that you can not have forces acting on it. 
A: By "static forces" I think you mean forces within a physical system in static equilibrium. 
You get static equilibrium out of the kinetic effects of force by noting the sum of the forces at every point is zero and therefore so is the sum of their kinetic effects.
