# I What Do Newton's Laws Say When Carefully Analysed

• Featured

#### bhobba

Mentor
Hi All

I will state the issue succinctly. This is a I level thread so I will suppose people know Newtons Laws. Newtons first law follows from the second which is a definition of force. So it has no actual testable physical content. The third law is equivalent to conservation of momentum as is proven in most texts on Classical Mechanics. This is not just a definition, but a testable statement about nature. However we know of this dandy theorem called Noether's Theorem, and this conservation law is equivalent to symmetry of spatial translation. But this is an assumed property of an inertial frame which is usually defined as a frame that obeys Newton's first law - but we have seen its not a law - but a consequence of the definition of force. Looked at it this way it seems a bit of a mess - yet its importance to many branches of science and engineering is beyond question. Can we disentangle this? (Of course we can - but this is just to set the stage to discuss it).

Thanks
Bill

Last edited:
Related Classical Physics News on Phys.org

#### russ_watters

Mentor
Can we disentangle this? (Of course we can - but this is just to set the stage to discuss it).
What is the desired output of this thread? E.G., what would a de-tangling look like? Just deleting Newton's 1st Law? Is it really necesary?

I do agree that Newton's 1st law is just a single point/special case of Newton's 2nd. And while that probably makes it superfluous, I think that's ok. Why do we have circles instead of just ellipses?

My guess - and I may be just inventing history here - is that since under Aristotelian physics objects slow down on their own (a sort of entropy?), it might have been necessary at the time to specify that force (N2), and only force (N1), will cause an object to accelerate/decelerate. So where today we may consider it obvious, at the time it may have needed to be stated.

#### A.T.

I do agree that Newton's 1st law is just a single point/special case of Newton's 2nd. And while that probably makes it superfluous, I think that's ok. Why do we have circles instead of just ellipses?
Regarding the other points raised by the OP:

You could also define force using Hooke's law, and use that to test the 2nd law.

The third law is more specific than conservation of momentum. For more than 2 bodies there are other possibilities to ensure conservation of momentum, than the 3rd law.

#### gmax137

I recall from previous similar threads that the 1st law ("an object in motion...") is a refutation of the prevailing (Aristotelian?) belief that motion requires a motive force, that an object in motion must be "propelled" by some agency. This archaic notion arises from observation of the everyday world where friction is always present. So I imagine that Newton starts with "first of all, that is just not true."

#### bhobba

Mentor
What is the desired output of this thread? E.G., what would a de-tangling look like? Just deleting Newton's 1st Law? Is it really necesary?
This thread originated on the QM forum from a post that mentioned some people take F=ma as a definition. There I mentioned the 'ultimate' answer to the issue is QM which is a bit surprising and interesting. Its likely of zero practical value, just for understanding purposes, and I think an interesting thing to work through.

Thanks
Bill

#### Mister T

Gold Member
Newtons first law follows from the second which is a definition of force.
This is a common misconception. In fact, many textbooks make the same error.

Newton's First Law is the assertion that being at rest is the same thing as moving in a straight line at a steady speed. It forms Einstein's 1st Postulate, that all inertial reference frames are equivalent. In other words, it's the Principle of Relativity.

So it has no actual testable physical content.
It's been very well tested experimentally.

The third law is equivalent to conservation of momentum as is proven in most texts on Classical Mechanics.
Not quite. The Third Law implies Conservation of Momentum, but Conservation of Momentum does not imply the Third Law. The Third Law is not valid on an instant by instant basis because forces take time to propagate. Modern physics has given primacy to Conservation of Momentum for this very reason.

#### bhobba

Mentor
You could also define force using Hooke's law, and use that to test the 2nd law.
Yes, but I do not think anyone has developed it that way.

The third law is more specific than conservation of momentum. For more than 2 bodies there are other possibilities to ensure conservation of momentum, than the 3rd law.
Good point. Its not equivalent to it.

Maybe there is a different way out. Despite force just being a definition could we look at it in a different way than we usually look at laws. Could we say its a prescription - that says get thee to the forces in analysing mechanical problems? I didn't come up with it - the resolution stumped me for years - its how John Baez resolved it in some discussions I had with him about it. It would then be different kind of law - not one that is testable but a statement on how to look at mechanical problems.

Thanks
Bill

#### russ_watters

Mentor
This is a common misconception. In fact, many textbooks make the same error.

Newton's First Law is the assertion that being at rest is the same thing as moving in a straight line at a steady speed. It forms Einstein's 1st Postulate, that all inertial reference frames are equivalent. In other words, it's the Principle of Relativity.
Is that really the historical context? Do you have any sources discussing that? I'm seeing a lot discussing the need for an explicit refution of Aristotelian physics, but then only the PIR as a secondary implication (or unstated prior assumption). If it's really just a statement of the PIR, why not just concisely state the PIR? Why the cumbersome wording?

#### DrStupid

Newton's First Law is the assertion that being at rest is the same thing as moving in a straight line at a steady speed.
Newton already used that equivalence in the definition of force some pages prior to the laws of motion. If he would have expected that to be important enough to make a special law for it, he would have done it before.

The first law is the qualitative definition of force as the cause for changes of the state of motion of a body - most probably as a differentiation from the Aristotelian force that was required to maintain motion .

#### russ_watters

Mentor
You could also define force using Hooke's law, and use that to test the 2nd law.
Agreed, so I think it needs to be worded to say that N2 is a description of force's relationship with motion (acceleration), not a general definition of force.
Yes, but I do not think anyone has developed it that way.
What's there to develop?

f=ma
f=kx

Why is one a definition of force and not the other? And of course:
f=GM1M2/r2

Calling one "the definition of force" seems....limited.

#### bhobba

Mentor
This is a common misconception. In fact, many textbooks make the same error.

Newton's First Law is the assertion that being at rest is the same thing as moving in a straight line at a steady speed. It forms Einstein's 1st Postulate, that all inertial reference frames are equivalent. In other words, it's the Principle of Relativity.
You got it. Just to restate it a bit differently. A inertial frame, by definition, is one where all points, instants of time, and directions are equivalent as far as the laws of physics are concerned. We know from experiment such frames to high degree of accuracy exist on earth. We now have apparatus such as modern atomic clocks that show all instants of time are not equivalent - or maybe just a Foucault pendulum . Certainly in intergalactic space we expect that to be inertial to an even higher degree of accuracy. It isn't hard to show that two inertial frames move at constant velocity to each other - nice little exercise. But that leaves an open question - are frames moving at constant velocity to inertial frames also inertial. That is an experimental issue and the answer is yes. In fact it has an even even higher symmetry - the laws of physics are the same in all inertial frames - which of course is the POR. Now lets consider an inertial frame with just a single particle at rest. If it shoots off in some direction then that would violate all directions being equivalent so it must remain at rest (I know its not strictly rigorous - but will do for now). So it remains at rest. This means if a single particle is moving then during an infinitesimal amount of time it is moving at constant velocity so we can go to a frame where it is at rest instantaneously. Since it is t rest in that frame it must remain at rest so in the original frame it must move at constant velocity. Particles that act the same as if they were the only particle in the system are called free - so we end up with free particles in an inertial system remain at rest or move with constant velocity ie the first law.

As you pointed out its a common mistake hardly ever talked about in texts. Landau does - but he is one of the few I know.

Now we just have the issue of the second law being a prescription rather than a usual law.

Any ideas on how to reformulate it as a more usual kind of law?

Not quite. The Third Law implies Conservation of Momentum, but Conservation of Momentum does not imply the Third Law. The Third Law is not valid on an instant by instant basis because forces take time to propagate. Modern physics has given primacy to Conservation of Momentum for this very reason.
Yes. But I think it is a hint how to reformulate it.

Thanks
Bill

#### bhobba

Mentor
Calling one "the definition of force" seems....limited.
Yes - is there a way that avoids the concept of force explicitly. Its not a trick question - just about any textbook beyond first year on classical mechanics develops it.

Thanks
Bill

#### Mister T

Gold Member
Is that really the historical context?
No. I was taking liberties. It took a long time and a lot of effort to go from what Newton and DeCarte thought about the so-called law of inertia to what Einstein thought of his 1st Postulate.

I think it better to say that Newton's First Law has stood the test of time, and has over time developed into the Principle of Relativity.

#### bhobba

Mentor
It took a long time and a lot of effort to go from what Newton and DeCarte thought about the so-called law of inertia to what Einstein thought of his 1st Postulate.
In practical terms for what we teach to engineers etc, and in solving problems (with a caveat I will mention later once its fully fleshed out) it is of no value at all. This is simply for those interested in a deeper understanding. I have mentioned the final outcome is its correct basis is QM - which is a surprising result. We are getting closer now to the logic of why that is. Just as a bit more of a hint - Feynman at first hated the resolution and even in graduate courses stuck to using forces. The irony is he was the one that first figured out its basis in QM, with some hints and help from a paper by Dirac. There I have nearly given it away.

BTW it took me ages to figure this all out even with hints John Baez gave me - it just goes to show how penetrating Einstein was to the heart of a problem. He, like only a few others, had a frightening ease with the substance behind the equations. Those few others were Feynman, Landau, and probably Fermi. But, as Feynman said, given what Einstein knew at the time he could not have done what Einstein did.

Thanks
Bill

Last edited:

#### vanhees71

Gold Member
It's not so easy. In every semester I've to teach Newtonian mechanics I'm struggling to get this complicated issue of the 3 postulates made clear, and I'm not sure whether I've found the perfect way teaching it.

I think you indeed need all three postulates to make a solid foundation, and one has to keep in mind that these postulates are not like a closed formal system of axioms as in mathematics but are the amalgamation of Newton's notion of fundamental observations needed to make his theory of mechanics, which starts as all physics with some notion of time and space.

For Newton time and space are absolute, i.e., independent of any physical process. They are just primitive qualitative observables. The trouble now is to make this notion of space and time quantitative. For Newton there was not so much choice. Space was without any doubt just Euclidean space, which indeed seems to describe the geometry of bodies in the real world, including a quantitative meausure for distance and angles.

Then you have time, and the observation of time is only possible through observing motions. To make that quantitative you need Newton's first postulate, which says already pretty complicated things. I usually state it as follows:

Newton I: It exists a frame of reference, consisting of an arbitrary fixed point in space and a (Cartesian) basis (three rigid perpendicular rods mounted at the fixed point), where for a non-rotating observer at rest a body stays at rest or in uniform linear motion if no forces change this state of motion.

Here already forces or their absence come in, and of course they are not defined, but it's observable, whether you are in an inertial frame. For this you need a body set in motion to define a quantitative measure for time. It's mapped to the distance the body moves along the straight line it's set in motion to and you define the time intervals, where the body moves by some given unit of distance along this line. Now you can check whether you are in an inertial frame, if all other bodies move in uniform linear motion relative to the "clock-setting" body.

After that you can discuss, how to physically determine an inertial frame. The best answer I can come up with is the local frame of a "fundamental observer" of cosmology defined as an observer who is at rest relative to the heat bath defined by the cosmic microwave background radiation, which you cannot really understand without GR and the cosmological standard model though.

Newon II: Now makes "force" quantitative by starting with defining the "change of the state of motion" as being proportional to the acceleration, which is well-defined given a measure of time and space intervals. Now not only force enters but also mass, and this is also worth a careful discussion. From a modern point of view mass in Newtonian physics is a quite complicated beast being a central charge of the Galilei Lie algebra in quantum mechanics ;-). Of course this you cannot make really clear in any way to beginners in the first mechanics lecture, but it's good to keep it in minde.

Newton III: is then not so probematic anymore.

Of course, another approach is to stop at Newton I, which establishes Galilei-Newton spacetime in a formal way, and then you can take recourse to the group-theoretical ansatz discussing the Hamilton principle first and then go with Noether, but that's also not an approach for the first mechanics lecture either, because it swamps the physics under a carpet of pretty complicated (though very beautiful) math.

At one point it's also good to watch the PSSC movie on "Frames of Reference", easily found on youtube.

Teaching the foundations is always the greatest challenge of all!

#### bhobba

Mentor
Of course, another approach is to stop at Newton I, which establishes Galilei-Newton spacetime in a formal way, and then you can take recourse to the group-theoretical ansatz discussing the Hamilton principle first and then go with Noether, but that's also not an approach for the first mechanics lecture either, because it swamps the physics under a carpet of pretty complicated (though very beautiful) math.
I have to give it to you - you basically said what I was getting at. Replace F=ma with the Principle Of Least Action and you get conservation of momentum and energy, including what they are from Noether.

I think once you have done Newtons 3 laws in a first year course (or at HS if you went to a good one) it's unavoidable beyond that it becomes more mathematical. I will give my reading list and order for a student who has done basic calculus at the end of the thread.

Now the question is - where does the Principle Of Least Action come from. You are on a roll Vanhees - how about you, as a particle physicist, explain that one. Then we are done.

Thanks
Bill

#### vanhees71

Gold Member
As usual the answer is a more comprehensive theory. In this case it's quantum theory and Feynman's path integral, which came from Feynman's curiosity, how the Lagrange formalism and the action (after all the fundamental natural constant entering the game through quantum mechanics is the Planck-action constant) is related with QT, and from a somewhat kryptic remark by Dirac in his famous textbook he came to the idea of the path integral, and there the least-action principle follows from the saddle-point approximation of the path integral for the propagator. For situations, where the typical values of the action are very large compared to $\hbar$ the saddle-point approximation is good and leads in leading order to the classical path given by the stationary point of the action. As it turns out that's just equivalent to the well-known WKB approximation of the Schrödinger equation, which also rests on a formal expansion in powers of $\hbar$ (which is an asymptotic rather than a real power series though since it's what the mathematicians call "singular perturbation theory"). This explains why in "classical situations", where the typical action is large compared to $\hbar$, the very path is the macroscopic trajectory of the system, for which the action is stationary, and why the quantum-statistical fluctuations around it are small.

#### olgerm

Gold Member
Newton 1. law is derivable from second (if $F=0$).

Newtons first law follows from the second which is a definition of force. So it has no actual testable physical content.
If accept force as intuitively understandable quantity, then newton's II law is not just definition of new variable $\vec{F}$, but stating a relation between 2 known quantities (acceleration $\vec{a}$ and force $\vec{F}$). If you accept that force is intuitively understandable quantity then newton 2. law is testable.

#### sophiecentaur

Gold Member
Newton 1. law is derivable from second (if F=0F=0F=0).
But it's more than that. An experiment to show Newton 2 will give a graph that 'near enough' passes through the origin; closer and closer as experiments get better. N1 is more than that and, of course, it was revolutionary when it ws first stated. It was a powerful argument to unify explanations of what goes on on Earth (where there is always friction, which told us that things always slow down) and what is observed in the Heavens, in which things keep going 'for ever'. So no 'driving power' was needed to explain orbits. A massive change in Science culture so you can't really leave it out and drop back to just two Laws.

#### bhobba

Mentor
Well that's it. The interpretation of the 3 laws is first clarifying what the first law means. The second law is a definition, but still has physical content when taken as a prescription in solving classical mechanics problems it says - get thee to the forces. The third law is testable and can be left as is.

However it is possible to get rid of law 2 and 3 and replace it with the principle of least action. This, unlike F=ma, is an actual statement about nature. Even more it satisfies one of the conditions of Noether's Theorem, and using it you can derive conservation of momentum, angular momentum and energy from the symmetry properties of inertial frames - which is the real content of law one - basically law 1 is the POR.

Another advantage is as the difficulty of the problems in classical mechanics rises, using forces to solve it becomes harder and the Lagrangian method easier. Feynman got caught in this as I mentioned before and insisted on using forces when Lagrangian's would have been much easier. Its ironic that one of his most famous discoveries was linking QM and Lagrangian's by his path integral version of QM.

Vanhees gave a good overview of it. Here is a not quite rigorous justification:

You start out with <x'|x> then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|......|xn><xn|x> dx1.....dxn. Now <xi|xi+1> = ci e^iSi so rearranging you get ∫.....∫c1....cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get ∫L dt.

Now Si depends on xi and Δxi. But for a path Δxi depends on the velocity vi = Δxi/Δti so its very reasonable to assume when it goes to the continuum L is a function of x and the velocity v. L is of course the Lagrangian.

In the classical world the paths very close to each other will only differ in phase so you can find a close path 180 degrees out of phase hence they both cancel. There is one exception however - when the path is stationary - here close paths are the same and you get reinforcement rather than cancellation. Hence one has, classically, the Principle of Least Action.

More detail can be found from a number of sources on the internet eg MIT:

My suggestion for reading after a course in calculus and a general calculus based physics course such as (I have a copy and like it because its cheap and not bad material wise - especially its relativity first approach):

Then Morin - Classical Mechanics (good treatment of relativity too):

After that I would get one of the most beautiful physics books I know - I fell in love with physics after reading it - before I was more into math - will not say anymore - the reviews on Amazon say it all - Landau - Mechanics

People often recommend Goldstein - but for me, Landau is THE book.

Just a note about the general physics textbook. There are a number about such as Fundamentals of Physics by Halliday and Resnick and Physics for Scientists and Engineers by Giancoli. As far as I can see all are good except for one you must be careful with. That's the famous Feynman Lectures on Physics. It's a masterpiece and recommended as a reference or supplement in many initial university level courses in Physics - but hardly ever as a main textbook for the course. As its co-author Mathew Sands said:

'It had always been clear that the Lectures, by themselves, could not serve as a textbook. Too many of the usual trappings of a textbook are missing: chapter summaries, worked-out illustrative examples, exercises for homework, and so forth...... I heard that most instructors did not consider the Lectures suitable for use in their classes, although some informed me that they used one or another of the volumes in an honors class or as a supplement to a regular text...... Most commonly, I was told that graduate students found the Lectures to be an excellent source of review for qualifying exams.'

It's pretty much a must have for your reference library - but not the best as an initial textbook.

Thanks
Bill

Last edited:

#### bhobba

Mentor
If accept force as intuitively understandable quantity, then newton's II law is not just definition of new variable $\vec{F}$, but stating a relation between 2 known quantities (acceleration $\vec{a}$ and force $\vec{F}$). If you accept that force is intuitively understandable quantity then newton 2. law is testable.
Of course we all have an intuitive idea of force. For example its harder to push or pull an object if it has more mass, and it doesn't matter what the object is, its the mass that is important. So we want our definition to conform to that. Suppose we have something that creates the same pull each time we use it. This could be a spring pulled the same distance. We have two objects of the same mass as determined by a mass balance. We apply the pull of the spring to one object and note the acceleration. Then we do it to the second object of the same mass - and find the same acceleration. Intuitively we think of the spring as giving the same force. So our experiment is consistent with F=ma or a=F/m. In fact doing this experiment we find, for the same force, a is proportional to 1/m or a=c/m for some constant c. This makes it very reasonable to define that c as something. Since we also find the c varies with the push or pull we apply ie depending in how the spring is compressed or pulled, it is reasonable to define c as a measure of the push or pull applied. We call it force to get F=ma. So yes the definition is consistent with our intuition, which is a good thing. But, and this is the key point - it is still a definition.

If you haven't read it you might like to read the chapter on what is force in the Feynman Lectures.

But what makes it a law of physics is that it should have some physical content. This is the bit that confused me for a long time and it was John Baez that finally gave me the answer. It's that, from experience, we find the essence of Newtons Laws is get thee to the forces. It not a normal law in the sense you can experimentally test it, but its reasonable to still consider it a law.

Thanks
Bill

Last edited:

#### atyy

Because the second law is a definition, the first 2 laws are empty unless supplemented by another law specifying the functional form of the forces that occur when objects interact, eg. F=GMm/r^2.

The 3rd law partially gives such content by constraining that functional form.

Once the 3rd law or another law like the gravitation law has filled the second law with physical content, then the first law inherits the content from the second law and these other laws, and together they define an inertial frame. In other words, an inertial frame is a frame in which all the laws of physics (not just the first, with F as a meaningless quantity not filled with the content of the third law and other laws like the gravitation law) are true.

Last edited:

#### Dr. Courtney

Gold Member
2018 Award
I'm still not convinced N2 is the best available definition of a force. It is the best available definition of inertial mass, if we can allow force to be defined in some other way. But the tension here illustrates that Newtonian Mechanics is not a logical system like Euclidean geometry where you have some definitions and axioms and go from there. Newtonian mechanics is more holistic and some of the ideas are necessarily circular. At best it is a self-consistent system. A force is a push or a pull. F = ma always applies to a NET force - the sum over all the forces present. It can only possibly serve as the definition of a single force if we have a compelling reason to believe only a single force is present. How does one make a case only a single force is present without referring back to N1 or N2?

But broaden out the system and conversation to include the universal law of gravitation, and you have a force, a force law relating force to motion, and plenty of empirical support that the system is both consistent and an excellent description of nature.

#### Stephen Tashi

Newton's First Law is the assertion that being at rest is the same thing as moving in a straight line at a steady speed.
What does "same thing" mean in this context?

#### gmax137

I'm still not convinced N2 is the best available definition of a force.
I'm not sure I understand the idea that N2 is a "definition." Is the idea, that "force" is hard to define, so we resort to "force is that which induces acceleration"?

That seems to ignore the common sense intuition of "push" or "pull." I know what it is, to apply force to the rear bumper of a stuck car, and I don't look to the car's motion to quantify "how hard" the force is I am applying. (This is, I think, what @Dr. Courtney is getting at by bringing "net" force into the discussion.) Now, you may object, that no, I don't "really know" what I mean by this force on the bumper. Trying to define it, and getting into circularities: push, pull, exertion, ... But many definitions are just so: collections of different words, that, reading through the list you eventually get the gist.

How about "time?" The dictionary says, "a nonspatial continuum that is measured in terms of events which succeed one another from past through present to future." Ok, what does "future" mean? "of, relating to, or constituting a verb tense expressive of time yet to come..." Circularities. And yet I have never seen "time" defined as "that parameter, which when position is differentiated twice with respect to it, provides the ratio between force and mass..." That might be some bizarre Jeopardy answer, in search of the question, "what is time" -- but a definition, sorry no.

I can see a definition when we write down $\vec v=\frac{d\vec S} {dt}$ or $\vec a = \frac{d^2\vec S}{dt^2}$

But I see a difference between that and , $\vec a = \frac{\vec F} {m}$

this last has physical content, it says something about real objects in the world.