I could prove that $d\vec{p}/dt = m\vec{a}$, but how can I prove that $\vec{F_{net}}=d\vec{p}/dt$? Is force just defined by this expression? What is the rigorous definition of a general force? In my textbook (matter and interactions vol.1), force is qualitatively defined as a "quantity of interaction".
Bonus:
How did Isaac Newton define "force"? How did he prove that $d\vec{p}/dt = \vec{F_{net}}$?
Equations of motions need not be proven: they are such because they are experimentally true and there is no basic reason for that (at this point I wonder how you proved that $\dot{\textbf{p}} = m \textbf{a}$: you must have done so just re-writing a different form of the same equation, or any other starting point which you assumed to hold true). In the universe where we live Newtons' laws hold (at least in same scales), but in principle it might also have been otherwise, with different laws and different equations. Physics does not investigate why some laws are true: it only investigates how we can describe such laws in terms of mathematical equations.
Newton did not define the force: he just realised that in inertial systems the only way to modify the velocity of a particle is to introduce an external function $F(x, \dot{x})$ doing the job. This is what we call the "Force" but in practical terms you never measure forces themselves, you only measure the impact they have on the motion, i. e. the acceleration. For example if you attach a massive particle to a spring, it will move according to the solutions of $$ -kx = m\ddot{x} $$ which you can measure directly in a laboratory. If true, you then define the force of a spring as $F(x) = -kx$.
I had this kind of question myself for a long time because before studying Physics I've studied Math and in Mathematics we do things quite differently. In Mathematics, a general procedure is to give some definitions, probably by specifying some axioms, then we derive and prove theorems from this.
On Physics there is a similar procedure, but it goes a little bit different. The idea is that based on observations we assume certain things. The justification for those things is that our observation sugested it, so there is not any logical reasoning behind that allows one to obtain those things from previously defined ones. Those things we assume, are like the axioms in Math. From those axioms, we derive results.
The results we derive will be true as long as what we assumed is true. So this provides a nice way to test those axioms. We go there and see if we observe our predictions. Provided they are observed this doesn't prove the axioms are in fact true, but gives us more confidence about it. Perhaps some times something will be observed which was not predicted and we revise those axioms.
Newton's laws are the "axioms" for Mechanics. Newton observed mechanical phenomena and based on observation assumed his three laws. In particular, the second law can be thought of as what force really is. Also, in Physics not always is possible to define everything in a precise way like in Math. So Force, for example, is defined as something which changes the state of motion of a particle. This is the best initial intuition on force and is what the second law tries to capture.
So, in summary, Newton defined force exactly as the change in the quantity of motion. Also, he did not prove that, he argued that should be assumed based on observation.
Law of inertia states an object in uniform motion continues in its state(i.e moving along straight line with uniform velocity) unless a force impressed thereon.. Mass is the measure of intertia of a body,i.e the massive the body greater the force required to change its state of motion... in the absence of ext.force the momentum (mv) of a object is unchanged.. But when an ext. force is applied we can guess a change in state of motion or change in velocity.. we can visualise that the greater the force the greater will be the change in velocity of the object(greater will be change in momentum).. similarly to move a object with a particular speed, heavier object will require more force.. from above two resasoning i can conclude that, whenever i see a change in momentum of a object i can be certain there is a ext.force impressed on it..
taking consideration the time of impact of force we can say if the change in momentum per unit time of the object is greater, then there MUST BE impressed a greater force.(because a small force for a long period of impact can produce greater change in momentum that a large force can do in small time of impact).. Since the change in momentum is a result of "Net External force" and its change per unit time is proportional to the fundamental concept Force..I can conclude "force as the chage of momentum of an object per unit time"... taking limit we can get the expression dp/dt=Fnet...
THIS ISN'T A PROOF BUT A FUNDAMENTAL APPROCH..
