Newton's second law is so intuitively obvious

[Andrew Mason]

That's a reasonable view of force - just not the way I would view it.

PLA - Principle Of Least Action.

Starting from that alone, and the the symmetries implied by Galilean Relativity, you can actually derive all of Classical Mechanics - strange but true - you will find the detail in Landau's book - Mechanics.

Landau starts with the principle of least action as a given. It may be interesting to do that but I am not sure it is science. You have to be able to make observations that the action is always a minimum, not make it an a priori assumption.

The principle of Galilean relativity is at least based on observation. And, by applying symmetry arguments, you can show that Newton's second and third laws of motion flow from that principle. And the principle of least action flows mathematically from the laws of motion. The importance that Landau accords to the principle of Galilean relativity indicates that he views it as a foundation for classical mechanics. I definitely agree with that.

AM

 

[T0mr]

Andrew Mason,

I am with you that the principle of least action feels like a priori assumption. However I do not see how Newton's Second Law, F=ma, can be derived directly from Galilean Relativity. I don't remember seeing that anywhere and I am curious.

 

[Andrew Mason]

Andrew Mason,

I am with you that the principle of least action feels like a priori assumption. However I do not see how Newton's Second Law, F=ma, can be derived directly from Galilean Relativity. I don't remember seeing that anywhere and I am curious.

See my post #20.

It is essentially a symmetry argument based on the first law and Galilean relativity (ie. absolute time and the principle that all inertial frames if reference (IFR) are equivalent: that the laws of motion are the same in all IFRs) and a certain concept of mass: that matter is made up of fundamental units of matter and that mass is simply a measure of the quantity or number of such units of matter.

A force F acting on a body m over a time interval δτ will cause a certain change in velocity of δv, as measured in the initial rest frame of m (IFR1). Moving now to the rest frame of m after the change in motion occurs (IFR2 ie. the IFR whose origin is moving at velocity δv with respect to IFR1) we measure the change in velocity of the same force F over a second identical time interval δτ. The equivalence of IFRs requires that there be the same change in velocity δv. If it were not the case, IFR1 and IFR2 would not be equivalent because the same force acting for the same time interval would effect different changes in motion. As measured in IFR1, the change in velocity is 2δv over time interval 2δτ. Repeating that you can see that Δv/Δt is constant. So a constant force causes constant acceleration.

Now, if you repeat the experiment by applying the same force to each of 2 identical bodies (each having the same quantity of matter) alternately for very brief intervals of δτ, the bodies accelerate at exactly half the rate in the first experiment. Because the force is only applied to them every other interval of δτ, it takes 2δτ to achieve a change in velocity of δv. So acceleration is inversely proportional to mass.

So that is pretty much the second law.

AM

 

[jhae2.718]

Those interested in this thread may find Truesdell's Essays in the History of Mechanics to be an interesting read.

 

[T0mr]

It is essentially a symmetry argument based on the first law and Galilean relativity (ie. absolute time and the principle that all inertial frames if reference (IFR) are equivalent: that the laws of motion are the same in all IFRs) and a certain concept of mass: that matter is made up of fundamental units of matter and that mass is simply a measure of the quantity or number of such units of matter.

I think I get for the most part what you are saying. All relative motions between IRFs would be the same because we have absolute time and space coordinates in Galilean Relativity. So as you have said each IRF would observe the other with the same speed but moving in the opposite direction which is the definition of relative velocity. I think I understand essentially what you are getting at by the example of alternating the infinitesimal amounts of time, δt, that acceleration is applied to each mass to effect a change, δv.

I think about force in a sort of similar way. I take it as something we defined to explain why the inertial quality of matter changes. I think scientists like Newton and Galileo understood through experiment that matter had an equilibrium state which was a motion at constant velocity (I take 0 to be a constant too). But they needed a concept to address when there was a change of that equilibrium state, or inertial motion, to another. That change takes place over a period of time and so it was appropriate call something that was a difference in the quantity of motion, mΔv, in some amount of time Δt the force.

Newton explains impressed force:
"Definition IV. An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertiae only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force."

So I definitely think its appropriate to look at inertial reference frames to help explain the origin and need for the concept of a force. And for the OP the concept of force may seem obvious now but I doubt at the time of its invention it was.

 

[stevendaryl]

A force F acting on a body m over a time interval δτ will cause a certain change in velocity of δv, as measured in the initial rest frame of m (IFR1). Moving now to the rest frame of m after the change in motion occurs (IFR2 ie. the IFR whose origin is moving at velocity δv with respect to IFR1) we measure the change in velocity of the same force F over a second identical time interval δτ. The equivalence of IFRs requires that there be the same change in velocity δv.

Okay, that's a nice argument, but it implicitly makes the assumption that a force has the same magnitude in all frames of reference. How is that justified?

 

[voko]

Landau starts with the principle of least action as a given. It may be interesting to do that but I am not sure it is science. You have to be able to make observations that the action is always a minimum, not make it an a priori assumption.

I do not believe it is a commonly accepted requirement that everything in a scientific theory must be directly observable. We usually require that something has to be observable, and if it is, it must be in agreement with an appropriate experiment.

Besides, your use of "always" is alarming. No observation or experiment could possibly attest that something is "always" so and so.

 

[harrylin]

That's a reasonable view of force - just not the way I would view it.

PLA - Principle Of Least Action.

Starting from that alone, and the the symmetries implied by Galilean Relativity, you can actually derive all of Classical Mechanics - strange but true - you will find the detail in Landau's book - Mechanics. Now for some shameless pithy admiration for that book:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
[..]

All of classical mechanics, in particular relating to force? If so, where is Hooke's force law?

 

[voko]

All of classical mechanics, in particular relating to force? If so, where is Hooke's force law?

Potential energy.

 

[A.T.]

Okay, that's a nice argument, but it implicitly makes the assumption that a force has the same magnitude in all frames of reference. How is that justified?

By assumption of Hooke's law? I guess you have to assume / postulate some parts.

 

[bhobba]

Hi Guys

Love the turn this thread had taken. I think that textbook by Landau is under-appreciated.

The relevance of the PLA is not that its subject to direct experimental verification, its that its required for Noethers Theorem to apply which gives the most elegant basis of CM. Its validity is most readily established from the axioms of Quantum Mechanics (its in tons of textbooks eg my go-to QM textbook Ballentine - QM - A Modern Development). Forces are definitely more intuitive and amenable to direct experimental verification - but beauty and elegance goes to the PLA version.

For what its worth I think its important to understand what's going on with both approaches. But as far as fundamental physics is concerned my nod goes to the PLA. The reason is that the PLA and Galilean Relativity leads to CM. The two axioms of QM (from which the PLA follows) and Galilean Relativity leads to Schrodinger equation etc ie non relativistic QM. The real essence of non relativistic physics is Galilean Relativity ie the POR and absolute time or, equivalently, an infinite speed effects can propagate - Landau is very careful to make clear that is foundational to the whole thing.

Thanks
Bill

 

[bhobba]

By assumption of Hooke's law? I guess you have to assume / postulate some parts.

Hooks law is an experimental observation regarding the behavior of springs - its not really a foundational law. Its not really a law because springs and similar things do not behave precisely that way - it more along the lines of Ohms Law. It tells us how resistors behave. Well what's a resistor - a device that to good approximation obeys Ohms law. So what's it's physical content? Its that things, we will call resistors, exist that to good approximation Ohms law applies to. And indeed from a simple model it can be justified. The same with Hooks law - its pretty easy to see the parallel.

Thanks
Bill

 

[voko]

Under-appreciated? At the third edition and in stock at Amazon? Hmm.

 

[bhobba]

Under-appreciated? At the third edition and in stock at Amazon? Hmm.

In terms of people discussing it.

I think Landau is well known to physics aficionados - at sort of a more advanced level of Feynman's Lectures.

I fell in love with Landau and physics in general from that book - like the shameless glowing review I quoted - it nearly made me weep when I first read it - it had that big an impact. I thought - this is what physics REALLY is about - beauty laid bare - like Euclid laid Geometry bare.

As you probably guessed I am more of the mathematical physics bent.

Thanks
Bill

 

[A.T.]

Hooks law is an experimental observation regarding the behavior of springs - its not really a foundational law. Its not really a law because springs and similar things do not behave precisely that way

All you need to assume is that the same force applied to the same object causes the same deformation in any frame. Since classically the size of objects is frame independent too, this leads to the frame independence of forces, which stevendaryl asked about.

 

[voko]

As you probably guessed I am more of the mathematical physics bent.

This is most interesting. In his original preface to the first Russian edition (1940, sans Lifshitz) he clearly delineated theoretical physics from mathematical physics. His book is on theoretical physics :)

 

[bhobba]

All you need to assume is that the same force applied to the same object causes the same deformation in any frame. Since classically the size of objects is frame independent too, this leads to the frame independence of forces, which stevendaryl asked about.

Yea - its very reasonable - just like Ohms law is very reasonable.

Thanks
Bill

 

[bhobba]

Your first post on Landau ("actually derive all of Classical Mechanics") implied that Landau derives Hooke's law. But I did not see that.

Sorry for any confusion. I did not mean to imply you can derive Hooks law from the PLA - you cant.

What you derive is the generalized force law. Landau assumes, for fundamental forces, it has the form of a conservative force. But in QM you can actually derive that as well.

Thanks
Bill

 

[voko]

Your first post on Landau ("actually derive all of Classical Mechanics") implied that Landau derives Hooke's law. But I did not see that. Perhaps I overlooked it. Or perhaps you slightly overestimate that book?

Granted, "derived all of classical mechanics" is an overstatement, to put it mildly. There is much in classical mechanics that he did not touch upon at all, including such venerable subjects as statics.

As far as Hooke's law is concerned, he actually derived it, without using the name, in the section on small oscillations, where he showed that any potential energy at small deviations from equilibrium is approximated by $\frac k 2 (q - q_0)^2$. Note he actually derived it, not postulated as is done in some other texts.

 

[bhobba]

As far as Hooke's law is concerned, he actually derived it, without using the name, in the section on small oscillations, where he showed that any potential energy at small deviations from equilibrium is approximated by $\frac k 2 (q - q_0)^2$. Note he actually derived it, not postulated as is done in some other texts.

Yea - that is the most elegant justification for it - its simply what's required for small deformations from a Taylor's expansion - derived - yes - but exactly where it starts to deviate from that approximation is an experimental matter.

Thanks
Bill

 

[bhobba]

This is most interesting. In his original preface to the first Russian edition (1940, sans Lifshitz) he clearly delineated theoretical physics from mathematical physics. His book is on theoretical physics :)

Interesting - just checked my well worn third edition and its not there.

Maybe I am more of a theoretical physicist - I don't know - my undergrad degree was in applied math and it just gelled so beautifully with that in my mind. At the root of a lot of math is symmetry - that physics was the same hit me like a thunderbolt.

Thanks
Bill

 

[harrylin]

Granted, "derived all of classical mechanics" is an overstatement, to put it mildly. There is much in classical mechanics that he did not touch upon at all, including such venerable subjects as statics.

As far as Hooke's law is concerned, he actually derived it, without using the name, in the section on small oscillations, where he showed that any potential energy at small deviations from equilibrium is approximated by $\frac k 2 (q - q_0)^2$. Note he actually derived it, not postulated as is done in some other texts.

Oh that's neat indeed!

 

[Andrew Mason]

Okay, that's a nice argument, but it implicitly makes the assumption that a force has the same magnitude in all frames of reference. How is that justified?

It is the same constant force that is seen to be constant in all frames. The force is constant because there is no change in the thing supplying the force. For example, a spring that is kept stretched to a constant distance; a very heavy weight that drops very slowly and supplies a force through a gear mechanism.

AM

 

[stevendaryl]

It is the same constant force that is seen to be constant in all frames. The force is constant because there is no change in the thing supplying the force. For example, a spring that is kept stretched to a constant distance; a very heavy weight that drops very slowly and supplies a force through a gear mechanism.

AM

But in Special Relativity, a force doesn't have the same magnitude in every reference frame. So to assume that it has the same magnitude in every reference frame is not an immediate consequence of the equivalence of all reference frames.

 

[Andrew Mason]

But in Special Relativity, a force doesn't have the same magnitude in every reference frame. So to assume that it has the same magnitude in every reference frame is not an immediate consequence of the equivalence of all reference frames.

But Galilean relativity does not apply in SR where time and space are not absolute. Those are premises in Galilean relativity.

In Galilean relativity all measurements of the thing supplying the force are identical in all IFRs while in SR they are not. For example, in Galilean relativity a spring stretched a certain distance as measured in one IFR is stretched the same distance in all IFRs. Not so in special relativity.

AM