Deduction of Newton's Third Law

Action and reaction forces are external forces when considering individual bodies.

However when we consider the bodies to form a system of bodies then the action and reaction forces become internal to the system.

One can then write for the forces acting on the system of bodies:

If we assume there to be no external forces acting on the system of bodies then:

Hence (1) simplifies to:

Newton' s second law tells us that only external forces can accelerate a sytem of bodies, ie., its center of mass.

Thus (3) reduces to:

From (4) it  follows immediately that each individual term   should also be zero, being independent.

Recalling that represents the action and reaction forces acting on the individual bodies of the system it follows that

each term has to consist of a pair of forces of equal magnitude  and directed in opposite directions.



Conclusison:  

We have shown above that

- Newton's third law is implicit in the second law

and that

- Forces can exist only in the form of pairs equal in magnitude and operating in opposite directions.

mandrin