# Newton's Third Law and conservation of momentum

I was looking through some reference material, and I came across this:

In the situations in which we cannot follow details, we need to know some general properties, that is, general theorems or principles which are consequences of Newton’s laws. One of these is the principle of conservation of energy, which was discussed in Chapter 4. Another is the principle of conservation of momentum, the subject of this chapter.

According to the text, energy and momentum are conserved as a consequence of Newton's Third Law.

But then I did some searching and it seemed that the conservation laws were much more fundamental than Newton's Third Law. The sources suggest that the action-reaction pairs are necessary for momentum to be conserved, so they are a consequence of the conservation of momentum.

Which is a consequence of which? Will it be right to treat one as a consequence of the other?

As far as the actual physics is concerned, it is meaningless to talk of whether conservation of momentum is "more fundamental" than Newton's third law -- you can axiomatise classical physics in either way -- from Newton's laws, from conservation laws, from symmetry laws, from an action principle, whatever. You can prove the resulting theories are equivalent, in the sense that all the alternative axiomatic systems imply each other.

In terms of understanding, it makes sense to have multiple different frameworks in your head -- a symmetry-based framework is really good intuitively, especially once you understand Noether's theorem, while an action principle is the most powerful and also more useful when you leave the realm of classical physics. Treating Newton's laws as axioms isn't a great idea -- it's mostly just historically relevant.

When you learn more advanced physics, conservation of momentum will start "feeling" more fundamental -- this is simply because momentum is an interesting quantity to talk about.

• Conservation of momentum is more fundamental than the third law of Newton, let's say, simply because it is always true while the third law is not. Oct 21 '18 at 21:34
• There is a reason why we talk more about momentum as we go ahead in physics. The reason is simply that momentum is a more fundamental quantity than force. It is a meaningful concept in generality whereas the concept of force is not. Oct 21 '18 at 21:35
• @DvijMankad I think what is more fundamental (if there must be) is symmetry. not laws of conservations. Oct 22 '18 at 19:40
• @Shing Yes, I agree that symmetries are quite fundamental. But with each continuous symmetry, there comes a conservation law. So, in the "table of fundamentalness", conservation laws arising out of symmetries are right there adjacent to the symmetries they are arising out of. But, more relevantly, my comments here deal with asserting what is more fundamental between the third law and the law of conservation of momentum. I never said (or am saying) that conservation laws are the most fundamental or are more fundamental than symmetries. Oct 22 '18 at 19:45
• @DvijMankad Fair point. My answer was mostly in reference to classical physics. Indeed conservation of momentum becomes the only possible (or at least best) way to phrase things in quantum mechanics. Oct 24 '18 at 10:06

Newton's third law can be regarded as an expression of the action principle plus translation invariance, assuming that all of the forces are sums of two-body forces (one body acting on one other body).

The action principle can be regarded as more fundamental than conservation laws, because conservation laws (like energy, momentum and angular momentum) can be derived from the action principle when the right symmetries are present (respectively time-translation, space-translation, and rotation symmetry). This is the subject of Noether's theorem. More than one theorem is named after Noether, but the one cited most often is the one I'm referring to here.

Another reason to regard the action principle as being more fundamental than conservation laws is that the action principle is still useful even when the usual conservation laws are trivial. A prominent example of this is general relativity, which respects the action principle but in which the usual symmetry-based definitions of energy and momentum are reduced to useless trivialities, essentially because in general relativity, translation symmetry is absorbed into a much richer kind of "local" symmetry called diffeomorphism invariance.

To be a little more explicit about what the action principle says mathematically, consider the context of Newtonian mechanics and suppose that the equations of motion for an $$N$$-particle system have the form $$m_n \ddot x_n(t) = F_n\big(x_1(t),\,x_2(t),\,...,\, x_N(t)\big)$$ where $$t$$ is time, $$x_n$$ is the location of the $$n$$th particle, $$m_n$$ is its mass, overhead dots denote time-derivatives, and $$F_n$$ is the force on the $$n$$th particle when the other particles are in the indicated locations. This equation is just a fancy version of $$F=ma$$ (written as $$ma=F$$). In this context, the action principle can be expressed by requiring that all of the forces $$F_n$$ may be written in terms of a single function $$V(x_1,\,x_2,\,...,\, x_N)$$ like this: $$F_n(x_1,\,x_2,\,...,\, x_N) = -\frac{\partial}{\partial x_n}V(x_1,\,x_2,\,...,\, x_N),$$ where $$\partial V/\partial x_n$$ means the gradient with respect to the location of the $$n$$th particle (which has $$D$$ components in $$D$$-dimensional space). By combining this with the assumption that $$V$$ is invariant under translations of all of the $$x_n$$ by the same amount, we can derive the conservation of momentum, for example.

Yes, in the light of the physics we know, conservation laws are more fundamental than Newton's third law of motion. The reasons being: the conservation laws are more generically valid than the third law of Newton, and (most of) the conservation laws are direct consequences of symmetries (the Noether theorems)--and as Weyl said,

"As far as I see all a priori statements in physics have their origin in symmetry."

Now, can we derive the third law of Newton as a consequence of the conservation of momentum? Yes, in the systems in which the third law of Newton is actually valid, we can derive it using the conservation of momentum. The systems in which the third law of Newton is valid are basically the systems in which momentum is carried only by material-particles (and the forces are two-body forces). In such systems, if a particle loses some momentum then some other particle must simultaneously gain the same amount of momentum for the momentum to be conserved. And thus, the rate of change of momentum of the first particle will be precisely the negative of that of the second particle--and thus, the third law of Newton.

So, it is perfectly fine to see the third law of Newton as a consequence of the more general law of conservation of momentum.

Can we think the other way around? No, not in general--because the law of conservation of momentum is generally valid while the third law of Newton is simply not valid in certain systems (such as the systems where fields also carry some momentum in addition to the momentum carried by particles). But, within the systems in which the third law of Newton is valid, we can very well see the conservation of momentum as a consequence of the third law of Newton. The proof of this statement is almost trivial: you can always find a pair-force to a given force such that the combined effect of the two forces on the composite system is no net change in momentum.

So, in conclusion, in generality, we should think of the conservation of momentum as a more fundamental law because it is more generically valid than the third law and is a direct consequence of the symmetry (of translations in space). But, in the contexts in which the third law of Newton is valid, either of the conservation of momentum or the third law of Newton can be seen as a consequence of the other.

In classical mechanics, the conservation of momentum follows from Newton's laws. Likewise in modern (special and general relativity) mechanics - conservation of momentum is integral to the rules of the game of mechanics.

But the conservation of energy is more fundamental; energy comes in many other forms than the ones described by mechanics: e.g. chemical energy, electrical energy, magnetic energy, nuclear energy - some of which were already old hat in Newton's time. And all of these are conserved, although they may be converted to other kinds of energy. And of course, energy can be converted to matter, and vice versa. But that is definitely post-Newton. And conservation of energy does not follow from any of Newton's laws, which are

1. Inertia, an object will remain at rest or in constant motion if not subject to an outside force
2. F=ma, force equals mass times acceleration, which is actually more a definition (of the novel physical concept "force") than a "law"
3. Action equals reaction (only valid in mechanics, not politics!), which is basically the law giving us the conservation of momentum

Actually,everything is inter connected and it is logical to think that something has to be first and others are mere consequences of it.

Pre-requisite knowledge:

• Let's talk about conservation of momentum:

Law of conservation of momentum is implied by newtons laws of motion and especially the third law.

• The concept of force itself came from the concept of 'momentum' i.e force is, but the rate at which momentum changes with time.

What the source says:

the conservation laws were much more fundamental than Newton's Third Law. The sources suggest that the action-reaction pairs are necessary for momentum to be conserved, so they are a consequence of the conservation of momentum.

How I understand that statement of the source:

You want to say that conservation of momentum is more basic than the concept of newtons third law.

My response to the statement:

I could,as well, say that 'conservation of momentum' is necessary for action-reaction pairs to exist

How all of this seems to me:

It's like the good-old: which came first,chicken or egg question?

My opinion:

What I'd like to say:

Since our concept of force has been derived from the concept of momentum I'd like to conclude that the concept of 'momentum' is more fundamental than that of 'force'.

How to intuitively understand what I just said:

• Assume that the concept of momentum doesn't exist:

What is force then?

• Assume that the concept of force doesn't exist:

what is momentum then?

well,I can still answer this question without having to know what force is.But, in the former case we couldn't answer what force is without having to know what momentum is.

But,which is a consequence of which : 'third law' is of 'conservation of momentum' or viceversa?

Does this statement:

well,I can still answer this question without having to know what force is.But, in the former case we couldn't answer what force is without having to know what momentum is.

imply that 'third law' is a consequence of'conservation of momentum'?

consider the following examples:

• Consider a single body of mass m moving in vacuum with a speed v.This body has mometum(P=mv).

Does the law of conservation of momentum hold true?

Yes,ofcourse because the momentum of a body remains constant.

where is newtons third law now?

• Let's say that you are observing a head-on elastic collision between two bodies,happening in vacuum.In this case momentum is conserved.The third law works.

Before collision we don't see any contact forces,so we don't know if newtons third law works or not,but there is a net momentum for the system and it is conserved.

During collision we know that newtons third law works and the system still has momentum conserved.

Now how can you to decide whether 'conservation of momentum' arose because 'newtons third law works or to 'conserve momentum' newtons third law works?

• In example 1 we saw that momentum is conserved for the body and we don't see any forces.

• In example 2: Imagine that you are testing the so called law of conservation of momentum.You observed that the bodies changed velocities after collision.you are trying to explain why that has happened because newtons first law says that a body cannot change its state of motion without the action of force.So you came up with the concept of contact forces and these forces follow the third law.

you also observed that net momentum of the system is same before and after collision

you want to know if the forces arose so as to conserve momentum or because the forces arose, momentum is conserved?

That's the: which came first,chicken or egg question?

why don't you think that these two concepts are two fundamental laws of nature?

i.e it's a property that when the two bodies interact,collide,there are equal and opposite forces and momentum is conserved before and after collision.

In our system of two bodies we observe that momentum is a property of a system and forces are interactions between two bodies of a system.

Conclusion:

Conservation of momentum is a property of the system and 'newtons third law'is a concept which describes the interaction between particles/systems.

You need one to explain the other in our case and most of the other cases too,both are interrelated.

you can use the concept of momentum to explain newtons third law and you can,also,use the concept of newtons third law and obtain the law of conservation of momentum

To put it simply:

we used the concept of momentum to understand force.After we understood it quite well we observed that when two bodies interact,like collision,there are two laws called'newtons third law' and 'conservation of momentum'.

Both,the egg and the chicken came first and last,as well.

NOTE:

I have given some bold statements in my answer.please correct me if you think that I am wrong anywhere