If one defines force as the time derivative of momentum, i.e. by F =d/dt 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?

Matias Aguirre 2022-07-17 Answered
If one defines force as the time derivative of momentum, i.e. by
$\stackrel{\to }{F}=\frac{d}{dt}\stackrel{\to }{p}$
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?
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Bianca Chung
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:
$\stackrel{\to }{F}=G\frac{Mm}{{r}^{2}}$
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 $\stackrel{\to }{F}$ in Newton's second law is the total force acting on a body having momentum $\stackrel{\to }{p}$. 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.