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Suppose that some frame satisfies Newton's second law, so that $F_{net} = ma$ for all particles in the frame. Does this imply that the frame satisfies Newton's first law? This seems to be the case, since if $F_{net} = 0$ and $m > 0$ then we must have $a = 0$.

I read some other answers on this site which state that we cannot define inertial frames using the second law. Said another way, it cannot be that the first law holds if and only if the second holds. If the above is true, this suggests that we can construct a frame in which the first law is satisfied but not the second. What's an example of such a frame?

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    $\begingroup$ You might want to link the answers you are talking about. $\endgroup$
    – Puk
    Oct 27 '20 at 3:56
  • $\begingroup$ @Puk For example: physics.stackexchange.com/q/122231. If we could define inertial frames as those frames which satisfy the second law, why would we need the first law? $\endgroup$ Oct 27 '20 at 3:56
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There are quite a few ways to interpret Newton's laws. Historically, his perspective was vastly different from our modern one.

One perspective would be to say that Newton's first law is meant to stand in contrast with the Aristotelian model of mechanics, in which forces determined a body's velocity, not acceleration. The second law is then a follow-up which explains how to compute the effects of the forces which act on the body.

A more modern perspective might be that the first law asserts the existence of an inertial frame of reference, while the second explains how to compute accelerations within that frame of reference. From an axiomatic point of view, the latter is useless without the former. If I tell you that all of my children have blonde hair, that doesn't give you much information about the universe unless I also assert that I have children in the first place - see vacuous truth for more about this logical technicality.

If you are asking questions about the logical independence of two statements (in this case, the first and second laws), then you are asking a question about the logical and mathematical content of the statements themselves which cannot be resolved by physical measurement. No measurement could rule out the possibility (i) that $\mathbf F = m\mathbf a$ holds in inertial frames, but (ii) that there are no inertial frames in the universe we occupy, and in our universe $\mathbf F = m(\mathbf a + \mathbf a_0)$, where $\mathbf a_0$ points toward Alpha Centauri with magnitude $|\mathbf a_0|=10^{-100} \frac{m}{s^2}$.

That may seem unlikely, but again this is a question of mathematical logic, not practical applications.

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  • $\begingroup$ You're saying that the purpose of the first law is to assert that inertial frames exist, so that it makes sense in the first place to build a theory on-top of them. Is that right? You also mentioned that there exist no real inertial frames in our universe, so they are purely theoretical. $\endgroup$ Oct 29 '20 at 1:00
  • $\begingroup$ Would you agree that this is analogous to studying the real numbers by listing the axioms a field satisfies and deriving results directly from the axioms, without working with an explicit construction? I want to make sure I’m understanding this correctly. $\endgroup$ Oct 29 '20 at 1:00
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    $\begingroup$ @TheProofIsTrivium My claim was that no empirical measurement (which is subject to a non-zero degree of experimental uncertainty) can demonstrate the existence of inertial reference frames, so their existence is ultimately a postulate of the theory (namely, Newton's 1st Law). The clearest analogy I can think of to eliminating Newton's 1st law is eliminating the axiom of the empty set from the ZF axioms. The other axioms may tell you how you could manipulate a set which has no elements, but that's not much good without the assertion that such an object actually exists in the first place. $\endgroup$
    – J. Murray
    Oct 29 '20 at 1:15
  • $\begingroup$ Thank you, that's a great answer. $\endgroup$ Oct 29 '20 at 1:29
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To put it simply, it is a (widespread) misunderstanding that the content of the first law of Newton is that a body continues to move at a constant velocity if it hasn't been subjected to an external force in an inertial frame. This is purely an unfortunate misunderstanding. Rather, this description is just the definition of an inertial frame, in particular, that an inertial frame is one in which a body continues to move with a constant velocity if it hasn't been acted upon by an external force.

The content of the first law of Newton is that inertial frames exist.

The logical independence of the second law of Newton and as to why the first law is not simply a special case of the second law becomes abundantly clear in this view. It is not that we cannot conclude from $F=ma$ that if $F=0$ then $a=0$ but that this is not the statement of the first law of Newton.

Adopting the widespread misunderstanding regarding the content of the first law of Newton also leads to some related misunderstandings such as thinking that the Newtonian laws of mechanics are circular, see, for example, Are Newton's "laws" of motion laws or definitions of force and mass?.

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As far as I can understand from here

Said another way, it cannot be that the first law holds if and only if the second hold.

Simply, you are saying the second law does not imply the first. These types of questions are raised by people from time to time, you can [see][1] [here][2], etc. I tried here the best analogy that I can give to make you understand.

Suppose there are two worlds.

  1. Non-inertial world denote it by $N$.
  2. Inertial world denote it by $I$.

A person $A$ wants to learn the mechanics of particles. The mechanics in both worlds are different from each other, So first He/she needs to know in which world he/she is? To do that He/she has a standard procedure. If a uniformly moving isolated body continues to move uniformly, then he/she confirm that He/she is in $I$ world. Now He/she can learn mechanics through the formula He/she learned means he/she can use $\mathbf{F}=m\mathbf{a}$ if He/she is in $I$ world.

Now suppose the procedure is a little bit change, He/she first applies $\mathbf{F}=m\mathbf{a}$ on the body, and from this he/she decides whether of not he/she in $I$ or $N$. Now suppose he/she measures a body accelerating with acceleration $\mathbf{a}$,then what next? How do he know it's right or wrong. He/she need to know what is the value of acceleration for the same body in $I$ world. But to evaluate that he/she need do all the procedure again. So you just trapped in there's only one way you can do the dynamic of particle and that is to know in which world you are in. You can not go other way around.

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Newton 1. and 2. laws in standard formulations are, in my opinion, quite confusing when you are trying to get deeper into them. I will use wikipedia formulation:

First law In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Second law In an inertial frame of reference, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F = ma. (It is assumed here that the mass m is constant – see below.)

Now, the question is what is this force that this laws refer to? And what is this inertial frame these laws refer to? Obviously they have to be defined independently of these laws. The best you can do is conjecture, that there is some vector quantity with certain properties, that captures the information about interactions between two bodies and that there is some inertial frame in which physics has some nice symmetries. The second law then tells you what kind of effect does this force has on movement of the bodies in this frame and first law is just consequence of the second.

In this formulation, you need to investigate motion of bodies to find out what kind of forces are there and what are these inertial frames. But the thing is, you cannot really define force in full generality without somehow mentioning the effect it has on a movement (and similarly with inertial frame). Only once you have specific formula, you can define force independently. For example, if you already have gravitational law, then you already have force which is not defined by the effect it has on movement of the bodies, but rather it is defined by the state of the bodies and the nature of the interaction. But if you do not have it and you only seek it, you cannot really say what is this force you are looking for without saying it should be some vector quantity that produces such and such movement. And you cannot say which frame is inertial without saying it is frame in which bodies behave according to the first law, even though if you have one already you can define it without reference to first law, you can say for example that inertial frame is rest frame wrt to distant stars.

Mathematically, you can say $F_{net}=0$ implies first newton law just by using our conjecture that there is some $F_{net}$ in an inertial frame. But physically, how do you know what is this $F_{net}$ and that it is zero? How do you know you are defining force in inertial frame? Thanks to first law, you do not need to know exactly what $F_{net}$ is. And it is good you do not, because you need inertial frame before you can wish to look for this $F_{net}$. You simply need confidence, that all the interactions are shielded in whatever frame. Now, intuitively, you can see when something is shielded. You are in vacuum, body is electrically neutral and so there should be no interaction. You make a hypothesis about some frame being inertial frame by observing in which frame the body is in straight uniform motion, then you seek forces in this frame, you will see there are some complications in your model, so you devise better shielding and iterate until perfect model is reached.

Simply put, the idea that you do not need to know what $F_{net}$ is before you start to look for inertial frame is so important it is well justified to put it in its own law, even though mathematically it is derivable from second one quite trivially.

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