Force and momentum, cause and effect, transition and state

Question

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?

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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.

Answer 1

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.

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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.

Answer 2

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.

Answer 3

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.