Controllability verifications of Newton's third law

Newton's third law of motion has two co-existing forms: the weak form and the strong form. The weak form asserts that the internals forces are equal (in magnitude) and opposite (in direction), while the strong form asserts the additional central (or collinear) condition. It has always been puzzling to the engineering and science community as to exactly which one is correct. This paper endeavors to resolve this discrepancy via control theory. It considers a problem of particle dynamics as a planar affine nonlinear system. The forces sustain particles motion, which consist of internal and external forces, may be viewed as functions of time and particles' motions. To verify if the internal forces are able to sustain the motion, a nonlinear controllability theorem is applied. We are able to deduce a definitive conclusion on which form of the Newton's third law is valid.