We know, that the momentum of a closed system is conserved and thus all of the internal forces in such system must add up to zero. But in most mechanics textbooks it is stated, that even if there are some external forces acting on the system, the internal forces also balance each other and thus we arrive at a very usefull result, that the total rate of change of the systems momentum, is equal to the net external force acting on the system. How can we justify such an assumption about internal forces? Why they can not depend on this external influences?
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1$\begingroup$ It is due to the third Newton's law. Sum internal forces over all particles and they cancel up in pairs. $\endgroup$– DiracologyMay 28, 2017 at 22:27
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Internal forces cancel. Always in classical mechanics. An object cannot move itself - byproduct of equal and opposite.
Mathematically the argument is something like this; true internal forces do exist, and yes they cause the mass components of the system to move. But were you to add all of these forces acting on their respective individual mass components (m) then the total integral would vanish to zero. This means that the CENTER of mass (M) does not move as a result of internal forces. Thus momentum of the object as a whole is conserved unless an external force is applied (which is capable of affecting the center of mass).
Imagine a blob of water in space. The water flows as inertia causes the molecules to hit and consequently have a force on one another. But the center of mass moves only according to inertia - it does not accelerate. Thus momentum of the TOTAL body is conserved. Some text are unclear about their definitions of momentum of mass components vs. momentum of center of mass.