F12 = –F21.
The other is the law of momentum conservation, expressed by
p1 + p2 = constant .
Is one of these laws more fundamental than the other? There is no unambiguous logical way to answer this question. However, physicists would generally rank momentum conservation as the deeper law because it remains a pillar of both relativity and quantum physics, whereas force does not. Originally Newton chose his three laws as the most suitable axioms for the development of mechanics (what we now call classical mechanics). He actually advanced the law of momentum conservation (in the absence of outside influences) as a “corollary” to his statement of the third law. Here is the way he stated it1:
The quantity of motion [momentum] . . . suffers no change from the action of bodies among themselves.
In other words, internal forces in a system sum to zero (because of the third law) and therefore cause no change in the momentum of the system.
Logically, momentum conservation for an isolated system could be adopted as a basic axiom of mechanics, in which case Newton’s third law would become a “corollary.”
Among the reasons for attaching a special significance to the law of momentum conservation is its simplicity. Every conservation law, by picking from the chaos of activity and change in nature a single constant quantity, has a special appeal. Because momentum is a constant for every isolated system and in every fundamental interaction, it is natural to regard it as a particularly important law.
As mathematicians of the eighteenth and nineteenth centuries studied and reformulated mechanics, the concept of force was eclipsed in importance by the concepts of energy and momentum. This was, so to speak, accidental. It simply turned out that the equations with the greatest power for solving mechanical problems were equations in which momentum and energy appeared explicitly, and force did not. Then momentum surfaced as a key concept in the new theories of the twentieth century, further overshadowing force.
What caused force to fade in importance? It encountered two main difficulties in modern physics. The first is that in the catastrophic events of annihilation and creation that govern the behavior of fundamental particles, we usually do not know what forces act. The conservation of momentum from before to after the event can be verified; the probability of the event can be determined; but exactly what forces are at work during the event (if indeed the event has any duration) cannot be measured. The second difficulty is more an inconvenience than a fundamental problem. It has to do with the idea of action at a distance. Since forces do not act instantaneously across space, but are propagated with finite speed, Newton’s third law must be examined with new care. We must ask: When are the pair of forces equal and opposite? It takes some time for particle number 1 to react to the presence of particle number 2 somewhere else, and in that time the distance between the particles and the force between them may have changed. The result of this is that Newton’s third law is greatly complicated, for it is no longer true that a pair of forces need be precisely equal and opposite at a given instant of time. On the other hand, the law of momentum conservation retains a simple form even when the time lag of propagating forces is taken into account. It is still true that the system has a constant total momentum at each instant of time. The only added feature of the new view is that only part of the total momentum is contributed by the matter in the system. The rest comes from photons or other ephemeral messengers that transmit force, energy, and momentum from one material part to another.
In some ways the most compelling of the reasons for regarding momentum conservation as a profound law of nature is the connection between this conservation law and the uniformity of space. The principle of the uniformity of space might be called the principle of the sameness of nothingness. How a law as general and important as the law of momentum conservation can rest on a principle apparently so innocuous is discussed in the following Essay. As emphasized there, the true beauty of every conservation law may reside in the symmetry principle upon which it is founded. The symmetries of space and time underlie the laws of conservation of energy, momentum, and angular momentum. There are subtler symmetries of nature underlying some other conservation laws.
1 From Andrew Motte’s 1846 translation.