# Force and momentum, cause and effect, transition and state

Dale, an experienced contributor to this site, offered a surprising explanation for Newton's postulate actio = reactio: In this answer he argued the "explanation [for equal but opposite forces] is the conservation of momentum".

I spontaneously took issue with that because the momentum of a system is state we measure. The law of its conservation means that it does not change across interactions of its constituents. The state is a result of these interactions, not vice versa. In my understanding we have local, "microscopic" interactions between fields and "matter" or, maybe, upon closer inspection, just between fields; and those interactions are of a symmetrical nature.

After my comment to Dale's answer I realize I should elaborate on the "cause and effect" from the title. When we consider an event like the collision of billiard balls or nuclear fission we have a lot of interaction going on that changes the momentum of the involved "parties". What we observe when we compare the system state S before and S' after this interaction is that the momentum is conserved; the sum of all changes is zero. The momentum in the state S' is a result of all changes that happened. That this momentum is equal to the momentum in the prior S is a consequence of the nature of these interactions which happened between S and S'. The symmetry of the interactions is the cause for the observed effect that the momentum change is zero.

To sum up, the symmetric nature of the interaction leads to certain constraints in the resulting state, most prominently the well-known conservation laws. These laws are emergent properties resulting from the peculiarities of the underlying interactions; nature doesn't "know" about momentum (or energy, or angular momentum etc.), and there is no mechanism that would allow abstract concepts to govern concrete interactions. (Of course, from an "anthropic" point of view these symmetries are essential for a stable universe; if interactions didn't preserve energy or momentum the universe would immediately self-destroy or disperse in runaway processes. But that's not a watchmaker fine-tuning the interactions, it's evolution.)

It's possible that my programming background lets me think too much in terms of state-transition diagrams which do not model nature that well: Obviously, interactions never really stop, and states are never really static. On the other hand, many interactions are fairly transition-like, from defenestrations to pair annihilation, so the model is not entirely off.

And I'm aware that the predictive or perhaps rather conceptual power of the abstract, emergent laws is enormous: The conservation laws as well as the thermodynamic laws have a huge impact on our understanding of the cosmos and help form new theories and gain new insights.

But the question remains: Can one in good faith say "forces oppose each other because of the fundamental axiom that momentum is conserved"1, instead of the other way around? Are the two sentences equivalent?

1 I would distinguish this statement from one I would readily subscribe to: "I would be really surprised if forces were asymmetrical because we are very convinced of the general principle that momentum be conserved across interactions. A counter-example would shatter physics as we know it (and likely explode or implode the universe)."

Because, apart from the anthropic argument, the conservation of momentum is not the reason or cause for anything — it is a consequence.

• Please do not substantially edit your question after it has received answers. This one isn’t too bad, but you really want to avoid edits to the question that invalidate the received answers. It makes the site a mess
– Dale
Sep 27 at 1:09

The symmetry of the interactions is the cause for the observed effect that the momentum change is zero.

There is no cause and effect relationship here. The symmetry and the conservation law have a logical relationship, not a causal one. Both are valid at all times and so neither precedes the other and thus neither can be said to be the cause of the other.

Similarly, the relationship between Newton’s 3rd law and the conservation of momentum is a logical and not a causal relationship. When Newton’s 3rd law is valid, so is the conservation of momentum (although the reverse is not always true).

Can one in good faith say "forces oppose each other because of the fundamental axiom that momentum is conserved", instead of the other way around?

Yes. From a logical perspective any logical framework consists of a set of statements that are true within the context of the framework. Very often you can take a subset of those statements and use them to derive the remainder. Usually we will call the chosen subset "axioms" and the remainder "theorems". Typically the axioms are considered "fundamental" while the theorems are considered "derived" ("emergent" is not a good description). However, it is common that you can choose a different subset to be the axioms and then what had previously been an axiom becomes a theorem instead. So there is generally quite a bit of room for choice in deciding which statements should be considered fundamental.

Because of this strong element of choice in deciding which statements should be considered fundamental, it is rarely fruitful to argue about it. However, since it is specifically my statement which is the source of the question, I will be glad to share the reasons why I choose to list conservation of momentum as the fundamental principle. It should be understood that other choices are possible and reasonable, so my answer does not exclude other contradictory answers from also being valid.

For me, the issue of choosing which statements to consider "fundamental" is based on parsimony and generality. I would like to have as few fundamental concepts as possible in order to explain experimental data, and I would like those concepts to apply as broadly as possible.

1. If we start with force as the fundamental concept then typically we define force through the 2nd law $$\Sigma \vec F = m \vec a$$, and momentum as $$\vec p = m \vec v$$. We can then use the 3rd law to derive the fact that $$\vec p$$ is constant for a system of objects interacting through Newtonian forces.

2. If we start with momentum as the fundamental concept then typically we define momentum as the conserved quantity associated with the spatial translation symmetry of the Lagrangian and force as $$\vec F = d\vec p/dt$$. We can then use the conservation of momentum to derive the fact that two interacting objects will have equal and opposite forces.

Thus far, they are largely equivalent: with both 1. and 2. we have force, momentum, conservation of momentum, and Newton's 3rd law. This works for mechanical forces like the normal force and friction, and it works for Newtonian gravity.

However, we run into trouble with 1. once we get to electromagnetism. In electromagnetism it is easy to get scenarios where two charges interacting with each other have non-equal-and-opposite forces. In that case, as a statement regarding the interaction of the charges with each other, Newton's 3rd law is violated. We can patch that by saying that each separately interacts with the EM fields, and so N3 works between each charge and the field. This is a reasonable patch, and since 2. assigns a momentum to the field anyway it is not as though you are losing anything compared to the other approach.

However, $$m\vec a$$ is not well defined for the field, so to use 1. does require some modification. Specifically, we can no longer define force by the 2nd law but have to define the force on the field in terms of the change in momentum of the field, $$\vec F=d\vec p/dt$$. And since $$m \vec v$$ is also not well defined for the field we have to modify the definition of $$\vec p$$ to be the conserved quantity associated with the spatial translation symmetry of the Lagrangian.

So at that point 1. already starts to look a lot like 2., and I simply switch to considering 2. as the fundamental one.

Furthermore, when you get to QM we encounter 3 body forces, for which I know of no patch for Newton's 3rd law. But the conservation of momentum still applies just fine.

So, in summary, 1. works for scenarios involving classical contact forces and Newtonian gravity. 2. works for scenarios involving classical contact forces, Newtonian gravity, electromagnetic forces, quantum mechanical interactions, and even general relativity. 1. can be adapted to work for electromagnetic forces but in doing so it starts to look a lot like 2. and as far as I know it cannot be adapted for the other scenarios. Therefore, because it is more general (2. works in scenarios where 1. does not) my preference is to consider 2. as fundamental, not the other way around.

• Thanks for the comparison of the two concepts under different interactions. Instructive. But I disagree with your idea that one can simply choose the set of axioms as one sees fit. That's a very (nice) mathematical idea; but physics is processes, not static relations. There is causality and the direction of time. During an interaction you have lots of $d\vec p/dt$ of all involved parties. You integrate them after the fact and lo and behold, their sum is 0. The interaction comes before the finding that the momentum is conserved, and is therefore the cause, not the effect. Sep 26 at 22:46
• Now I realize that the symmetry of interactions and the symmetry (or other constraints) of the resulting states are perhaps, in a way, two sides of the same medal. Perhaps I'm asking a pseudo question. Not sure. Sep 26 at 22:50
• @Peter-ReinstateMonica causality is not relevant here. The relationship between N3 and CoM is not a causal relationship. It is a logical one. As such it can indeed be looked at either way. The temporal order that you propose is not correct. When two objects are interacting classically both N3 and the CoM hold at all times. There is no instant when one holds and the other does not. And no integration is needed. Describing one as cause and the other as effect is incorrect
– Dale
Sep 27 at 0:16
• Similarly for the translation symmetry. The symmetry and the CoM have a logical relationship, not a causal relationship. They are both always valid. One is not before the other in time
– Dale
Sep 27 at 1:12
• Oh, the wikipedia article is interesting -- funny enough, it didn't occur to me that there may be one. And it addresses exactly the question, did you write that yesterday? ;-) Jokes aside: Thanks. Sep 28 at 14:55

Can one in good faith say "forces oppose each other because of the fundamental axiom that momentum is conserved", instead of the other way around? Are the two sentences equivalent?

That momentum is conserved can be derived from Newton's laws of motion. That Newton's third law holds can be derived from conservation of momentum if one assumes that forces are pairwise interactions and that forces are instantaneous. So in that sense, they are equivalent.

But what if Newton's laws of motion are violated, which they observationally are violated in electrodynamics, in relativity theory, in quantum mechanics, and in thermodynamics? Yet conservation of momentum (or the relativistic equivalent) still holds by all experimental tests. Conservation of momentum is more fundamental than is Newton's third law.

• Well, the conservation is still a consequence of the peculiar interactions, right? It's just that one cannot conveniently ignore fields, or that the distinction between "particle" and "field" is a misconception. I admit that "force" is probably a misconception as well, on the microscopic/quantum level, so we have to use a more generalized concept of interaction, but still. The "entities" (fields, particles etc.) do not interact the way they do because they need to obey the law of the conservation of momentum. Obviously. It's the other way around. Sep 26 at 15:03

Can one in good faith say "forces oppose each other because of the fundamental axiom that momentum is conserved"1, instead of the other way around? Are the two sentences equivalent?

Consider collision of pair billiard balls, from conservation of momentum follows that : $$p_{tot} = \text{const} ~~~~~~~~~~~~~~~(1)$$

From $$(1)$$ follows, $$\frac {dp_1}{dt} + \frac {dp_2}{dt} = 0 ,~~~~~~~(2)$$

i.e., if total change in momentum before and after interaction would not be zero, then $$(1)$$ would be invalid.

Now move second billiard ball momentum change to the right hand side of equation :

$$\frac {dp_1}{dt} = - \frac {dp_2}{dt} ~~~~~~~~~~~~(3)$$

Then we know that $$F=\dot p$$, hence we can restate $$(3)$$ equation as:

$$\textbf F_1 = - \textbf F_2$$

So we derived third Newton law from conservation of momentum backwards.

• You could do the same arithmetic transformation with a relation between calorie intake and body masses M1 and M2 before and after; but few people would argue that the explanation for the eating was the body mass M2: It's the other way around. You can do that in pure mathematics where everything is static relations with no prejudice to either side. You cannot always do that in physics which describes systems that are dynamic in time. It's still mathematically correct as long as both sides are universally valid, but it's confusing cause and effect. Sep 27 at 9:46
• As others have mentioned, the relationship between these laws is a logical one and not causal, so do not inject additional meaning where there is no one. It's just a correlation between the third Newton law and conservation of momentum, which works both ways. You can explain one in terms of the other. (Read answer of Dale). Sep 27 at 10:51
• I read his answer, rest assured. My argument concerned your answer though. I gave an example where arithmetic equivalences are not a valid argument for conflating cause and effect. You don't say in your post what distinguishes force-momentum relationships from other relations where arithmetic equivalence obviously does not show logical equivalence. Of course you can refer to Dale's post for an argument, but since the arithmetic was never in doubt showing it does not add that much insight beyond Dale's answer, does it? Sep 27 at 11:22
• This arithmetic proof shows that you can logically descend from one law to the other. In contrary to what you have said,- this arithmetic proof shows logical equivalence between the laws, because arithmetic is simply one form of logic. (Albeit this proof has nothing to do with cause and effect relationships). Sep 27 at 11:29
• Explanations does not necessarily have to include a causal relationships,- there can be also an explanations which shows correlation between something, as in this case - correlation between physics laws. IMHO, causal relationships should target physical phenomenon, not laws itself. Like you add $N$ neutrons into a nuclei and the effect will follow - nuclei will break apart. But logical relationships between laws is just pure logics, transformation rules from one to the other. Physics law/equation on it's own does not state any causal relationships, just because it can be expressed either way. Sep 27 at 12:40