Newton's Third Law of Motion

Newton's third law of motion is

To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts.

Consider a dynamical system consisting of two bodies, labelled 1 and 2. Let body 1 exert a force ${\bf f}_{21}$ on body 2, and let body 2 exert a force ${\bf f}_{12}$ on body 1. According to Newton's third law of motion,

$${\bf f}_{12}= - {\bf f}_{21}.$$ (1.21)

In other words, the two forces are equal and opposite. In Newtonian language, one of the forces is the action, and the other is the reaction. Thus, action and reaction are always equal and opposite. Newton's third law holds irrespective of the nature of the forces acting between the two bodies. One corollary of this law is that a body cannot exert a force on itself. Another corollary is that all (non-fictitious) forces in the universe have corresponding reactions.

It should be noted that Newton's third law implies action at a distance. In other words, if the force that body $1$ exerts on body $2$ suddenly changes then Newton's third law demands that there must be an immediate change in the force that body $2$ exerts on body $1$. Moreover, this must be the case irrespective of the distance between the two bodies. However, we know that Einstein's special theory of relativity forbids information from traveling through the universe faster than the speed of light in vacuum. (See Section 3.2.10.) Hence, action at a distance is also forbidden. In other words, if the force that body $1$ exerts on body $2$ suddenly changes then there must be a time delay, which is at least as long as it takes a light ray to propagate between the two bodies, before the force that body $2$ exerts on body $1$ can respond. Of course, this means that Newton's third law is not, strictly speaking, correct. However, as long as we restrict our investigations to the motions of dynamical systems on timescales that are long compared to the time required for light-rays to traverse these systems, Newton's third law can be regarded as being approximately correct.