# Newton's Laws: Tautological?

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1. Jan 25, 2016

### Cosmophile

Hey, all. I've been studying Newton's Laws and I can't help but to feel a bit uneasy as I inspect them. From my point of view, they seem to be tautological. I'll refer to Newton's First Law as N1L, and similarly for the other two

Right off the bat, I suppose I don't see any reason for N1L to exist, at least not as a "law." It's certainly a good axiom from which the framework of Newtonian mechanics must be built upon, but a "law?" The information from N1L is easily derived from N2L (as is the information from N3L, as far as I can tell). N1L seems to just be a restatement of N2L, hence my claim for tautology.

If, as Kleppner & Kolenkow say, forces arise from interactions between systems, then N3L is just an explicit statement of this fact -- but is it a necessary one? If system $A$ interacts with system $B$ and therefore exerts a force $\vec{F}_{AB}$ on $B$, then $B$ also exerts a force $\vec{F}_{BA} = -\vec{F}_{AB}$ on $A$, which (if I am correct) can be easily justified from N2L.

Also, as a sidenote: K&K say that forces arise from interactions between system. I've thought about this a great deal and have to ask: would it be more correct to say forces are the means by which systems interact and that accelerations arise from interactions between systems?

Sorry for asking such tedious questions, but I've been both bothered and fascinated by these for the past two days and would love to discuss it with you all and hear your insights!

2. Jan 25, 2016

### axmls

No, a force doesn't have to correspond to an acceleration. See: statics--the study of systems where $\sum F = 0 \implies a = 0$.

Newton's first law is important because it establishes what an inertial reference frame is. This is important because Newton's second law is kind of intended for inertial reference frames (I believe). If you try to apply it in noninertial reference frames, you have to throw in fictitious forces (like the centrifugal force pushing you to the side when you make a sharp turn in your car). In essence, those fictitious forces are side effects of being in a noninertial frame.

3. Jan 25, 2016

### mfig

Eddington seems to agree with you on N1L. He says this essentially boils down to a statement that, "...every particle continues in its state of rest or uniform motion in a straight line except insofar that it doesn't."

Marion and Thornton call the first two laws mere definitions, and refer to the 3rd law as the true physical law. They seem to define physical law as:

A mathematical summary of near universal experimental fact, whose mathematical implications have also been experimentally verified.

They do concede that there is room for interpretation here, however. Other authors have considered N2L to be the physical law and the others as definitions.

4. Jan 25, 2016

### Cosmophile

Sorry, but isn't the above just an application of N3L? I'm not too familiar with statics, but if $\sum \vec{F} = 0$, doesn't it just mean that all forces acting on a mass are being canceled by forces being applied in the opposite direction? That is, I'd say (and I may very well be wrong about this!) that forces do correspond to accelerations. That is, for every force acting on a mass, there is a corresponding acceleration done by that mass. It may very well be the case that there are multiple forces acting on a mass, and that those forces could cancel out so that $\sum \vec{F} = 0 \quad \therefore \quad \sum \vec{a} = 0$.

Right, I agree with you that N1L establishes what an inertial reference frame is, and I agree with its importance. As I said, my issue with N1L is that we call it a "law," when I think it would be better recognized as an axiom of mechanics.

Interesting. I see N2L as being a physical law, with N3L being a consequence of N2L and a difference of reference frames. Again, I could definitely be wrong here.

5. Jan 26, 2016

### A.T.

How would you derive N3L from N2L?

6. Jan 26, 2016

### Cosmophile

We know that $\vec{F} = \frac{d \vec{p}}{dt}$. Consider two bodies $A$ and $B$, both having constant mass, which exert forces $F_{AB}$ and $F_{BA}$ respectively upon one another. From the conservation of momentum, we know:

$$\frac{d \vec{p}_{total}}{dt} = \frac{d \vec{p}_A}{dt} + \frac {d \vec{p}_B}{dt} = \frac {d(m_A \vec{v}_A)}{dt} + \frac {d(m_B \vec{v}_B)}{dt} = 0$$

Because $A$ and $B$ have constant masses, we can rewrite the above to say:

$$m_A \frac {d\vec {v}_A}{dt} + m_B \frac {d \vec{v}_B}{dt} = m_A \vec{a}_A + m_B \vec{a}_B = \vec{F}_A + \vec{F}_B = 0$$
$$\therefore \vec{F_A} = -\vec {F_B}$$

7. Jan 26, 2016

### Staff: Mentor

How do you know that momentum is conserved? In Newtonian mechanics that is usually derived from Newton's third law, so assuming conservation of momentum is the same as assuming Newton's third.

Personally, I regard Newton's first as a definition of inertial frames, and Newton's second as a definition of force, so those two are tautological. Newton's third is a physical law that introduces conservation of momentum.

8. Jan 26, 2016

### weirdoguy

Which is derived from N3L.

9. Jan 26, 2016

### Andy Resnick

They are tautological, in the sense that N2 is a definition. Arons' excellent book (https://www.amazon.com/Teaching-Introductory-Physics-Arnold-Arons/dp/0471137073) devotes a whole chapter to the topic, and discusses Mach's alternative presentation (starting with 'mass' rather than 'force').

As for the usual 'Newtonian' presentation, N2 must be accompanied by specific force laws (F = kx, Gmm'/r^2, etc) which cannot be obtained through N2.

10. Jan 26, 2016

### andresB

As the above quote from Eddington imply, the 1st law doesn't say anything by itself. The 2nd law also asy little without the acknowledgment of inertial systems.

IMHO, Newton laws are not independent from each others. they are not like a true mathematical axioms and I don't see anything wrong with that. They, as a whole, transmit a clear message and that is what matter.

11. Jan 27, 2016

### Cosmophile

Interesting points about CoM and N3L...I'll give this some more thought. If it's okay, I'd like to request that this thread be left open (I'm not sure what the conditions are for closing threads).

12. Feb 14, 2016

### A.T.

1) As other noted, N2L doesn't imply conservation of momentum, so you cannot use that.

2) Even assuming conservation of momentum, you cannot derive N3L. Your derivation works only for two bodies, not in general.