What Do Newton's Laws Say When Carefully Analysed

[DrStupid]

A non-interacting object has no external force.

Where does that come from?

PS: I just see that there are a lot of new post about this topic. Are they sufficient to explain why Newton III is required to define inertial frames with the laws of motion or does somebody still think it is possible with Newton I only?

 

[Dale]

Where does that come from?

A force comes from some interaction. No interaction -> no force. That is a pretty commonly used fact in drawing free body diagrams and setting up problems. Sometimes a problem will call such an object "isolated".

PS: I just see that there are a lot of new post about this topic. Are they sufficient to explain why Newton III is required to define inertial frames with the laws of motion or does somebody still think it is possible with Newton I only?

N3 is not required. N1 is sufficient for defining an inertial frame.

 

[DrStupid]

A force comes from some interaction.

Again, where does that come from if not from Newton III?

That is a pretty commonly used fact in drawing free body diagrams and setting up problems.

Fictitious forces are also commonly used.

N3 is not required. N1 is sufficient for defining an inertial frame.

Show me how to do that. Step by step, without conditions that fall from the sky.

 

[Jimster41]

You start with one non-interacting particle running along a straight line. If it doesn't go along a straight line, you already know that your reference frame is not inertial,

How is this established?

 

[Dale]

Again, where does that come from if not from Newton III?

No. We haven’t defined forces yet. We are not making any claims about what happens when objects do interact. We are simply starting with non-interacting objects and stating that their motion defines inertial reference frames.

Show me how to do that. Step by step, without conditions that fall from the sky.

I already did. I am not sure what conditions you are considering falling from the sky or what steps you feel are missing.

 

[vanhees71]

Show me how to do that. Step by step, without conditions that fall from the sky.

I did so three times already in this thread. What do you consider not to be sufficient? Of course, you cannot establish the laws without taking empirical input, which are indeed in a sense "conditions that fall from the sky". Newton's genius lies in his intuitive choice of the empirical input in terms of Newton I-III.

Of course, you need all 3 postulates to get Newtonian mechanics as a whole. To define an inertial reference frame you only need Newton I.

 

[vanhees71]

How is this established?

By observing, whether it runs along a straight line. The entire notion of space as an Euclidean affine space is assumed by Newton without explicitly mentioning it, because for him there were no other geometries available. Thus the notion of straight lines, dinstance, angles, and circles, are all tacitly assumed by Newton.

 

[DrStupid]

No. We haven’t defined forces yet. We are not making any claims about what happens when objects do interact. We are simply starting with non-interacting objects and stating that their motion defines inertial reference frames.

You claimed above:

N3 is not required. N1 is sufficient for defining an inertial frame.

Instead you again start with "non-interacting objects". Which part of N1 does that come from? Citation please!

I am not sure what conditions you are considering falling from the sky or what steps you feel are missing.

This one:

A non-interacting object has no external force.

That is a consequence of N3. You are claiming not to use N3, but you failed to explain where it comes from instead.

 

[Herman Trivilino]

@DrStupid : Law III describes interactions, so you don't need to invoke it unless you have interactions.

You don't need the rules describing how to put frosting on a cake if you want to make a cake that has no frosting.

 

[DrStupid]

What do you consider not to be sufficient?

Newton I and II are not limited to interactive forces without Newton III because Newton III and only Newton III excludes fictitious forces. Without this limitation to interactive forces there is no limitation to inertial frames.

 

[DrStupid]

Law III describes interactions, so you don't need to invoke it unless you have interactions.

Newton III not only describes interactions. It says that interactions are required for forces:

"Actioni contrariam semper et aequalem esse reactionem"
(To every action there is always opposed and equal reaction)

No interaction would not mean no force without this requirement. But that is the basis for the derivations of inertial frames from Newton I proposed in this thread. Therefore they do not work without Newton III.

 

[gmax137]

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.

Is this idea developed anywhere? I think I understand the instant by instant idea, but when I try to work it out, my head starts spinning.

I think we did this in intro EM (Purcell); does that sound right? Moving charges, that stuff. It has been decades since I gave EM any thought.

Is instantaneous action considered to be another "built in" feature of Newton's classical mechanics (like absolute time & space) and thus, kind of outside the scope of this thread?

Thanks

 

[kith]

No interaction would not mean no force without this requirement. But that is the basis for the derivations of inertial frames from Newton I proposed in this thread.

What if @vanhees71 replaced "non-interacting particles" by "force-free particles" in his post #57? It seems to me that you would still object but what exactly would your objection be then?

 

[kith]

Is this idea developed anywhere? I think I understand the instant by instant idea, but when I try to work it out, my head starts spinning.

If you take the second law as the definition of force, it is defined as the rate of momentum change. The third law than says that for two interacting bodies, their rates of momentum change are always opposite but equal.

If our interaction is not instantaneous but mediated by a field, that's not true on an instant by instant basis anymore because the field carries non-zero momentum at intermediate times. I.e. the momentum gain of one of the bodies happens later than the momentum loss of the other.

 

[Swamp Thing]

If our interaction is not instantaneous but mediated by a field, that's not true on an instant by instant basis anymore because the field carries non-zero momentum at intermediate times. I.e. the momentum gain of one of the bodies happens later than the momentum loss of the other.

If we are willing to attribute momentum to the field or EM wave, then it does apply on an instant by instant basis, yes? We can't always assume that there is an object somewhere out there waiting to catch the field momentum (maybe a few light years away) and thus balance the books -- so we would have to count the field momentum as part of the conservation balance, right?

 

[Dale]

Instead you again start with "non-interacting objects". Which part of N1 does that come from? Citation please!

Chapter 1 of Lagrangian and Hamiltonian Mechanics by MG Calkin begins "Newton's first law deals with non-interacting bodies. It says that the velocity of an isolated body, one removed from the influence of other bodies, is constant. This law defines a set of preferred coordinate frames, inertial frames, as frames in which Newton's first law holds." https://www.worldscientific.com/doi/pdf/10.1142/9789810248154_0001

This is by no means a personal formulation of mine or @vanhees71, it is an accepted meaning of N1.

That is a consequence of N3. You are claiming not to use N3, but you failed to explain where it comes from instead.

Completely disagree. N3 doesn't even apply to non-interacting bodies. It says nothing whatsoever about them. An object is either interacting or non-interacting. If it is non-interacting then N1 defines inertial reference frames such that non-interacting objects have constant velocity. If it is interacting then N3 describes how pairs of interacting objects exert forces on each other. If it is interacting then N1 is silent and if it is non-interacting then N3 is silent.

N1 is not a consequence of N3. They are not even applicable to the same bodies, so I don't see how you can possibly think it is relevant. N1 starts with non-interacting bodies and uses them to define an inertial frame. As such, it is simply a definition and is not a consequence of anything. Definitions don't have to "come from" anything.

 

[kith]

If we are willing to attribute momentum to the field or EM wave, then it does apply on an instant by instant basis, yes?

By "it", you seem to mean momentum conservation and yes, this applies on an instant by instant basis. But we were talking about Newton's third law which implies that momentum is exchanged instantly between bodies. This is false in elaborate theories of interactions like electromagnetism where fields mediate the interaction. I consider this to be a minor point. The important thing is that there's a general principle which holds for all physical systems: momentum conservation.

We can't always assume that there is an object somewhere out there waiting to catch the field momentum (maybe a few light years away) and thus balance the books -- so we would have to count the field momentum as part of the conservation balance, right?

Sure. It's just that if there's a body which catches the momentum it can't happen at the same instant as another body gives it to the field.

 

[kith]

Griffiths also gives a general argument against the third law in relativistic theories in his "Introduction to Electrodynamics" (see chapter 12.2.4 "Relativistic dynamics"):
"Unlike the first two, Newton’s third law does not, in general, extend to the relativistic domain. Indeed, if the two objects in question are separated in space, the third law is incompatible with the relativity of simultaneity. For suppose the force of $A$ on $B$ at some instant $t$ is $F(t)$, and the force of $B$ on $A$ at the same instant is $−F(t)$; then the third law applies, in this reference frame. But a moving observer will report that these equal and opposite forces occurred at different times; in his system, therefore, the third law is violated."

 

[gmax137]

therefore

this is why i asked

Is instantaneous action considered to be another "built in" feature of Newton's classical mechanics (like absolute time & space) and thus, kind of outside the scope of this thread?

 

[atyy]

Interesting history of the concept of inertial frame: https://plato.stanford.edu/entries/spacetime-iframes/

"Neumann's definition of a time-scale directly inspired Ludwig Lange's conception of “inertial system,” introduced in 1885 . An inertial coordinate system ought to be one in which free particles move in straight lines. But any trajectory may be stipulated to be rectilinear, and a coordinate system can always be constructed in which it is rectilinear. And so, as in the case of the time-scale, we cannot adequately define an inertial system by the motion of one particle. Indeed, for any two particles moving anyhow, a coordinate system may be found in which both their trajectories are rectilinear. So far the claim that either particle, or some third particle, is moving in a straight line may be said to be a matter of convention. We must define an inertial system as one in which at least three non-collinear free particles move in noncoplanar straight lines; then we can state the law of inertia as the claim that, relative to an inertial system so defined, the motion of any fourth particle, or arbitrarily many particles, will be rectilinear. ... "

"... More simply, an inertial reference-frame is one in which Newton's second law is satisfied, so that every acceleration corresponds to an impressed force. Thomson did not reject the term “absolute rotation,” holding instead that it has to be understood as rotation relative to a reference frame that satisfies his definition. The definition does not express, as Lange's does, the degree of arbitrariness involved in the construction of an inertial system by means of free particles. Moreover, like Lange's, it leaves out a crucial condition for an inertial system as we understand it: all forces must belong to action-reaction pairs. Otherwise we could have, as on a rotating sphere, merely apparent (centrifugal) forces that are, by definition, proportional to mass and acceleration, and so the rotating sphere would satisfy Thomson's definition. Therefore the definition needs to be completed by the stipulation that to every action there is an equal and opposite reaction. (This completion was actually proposed by R.F. Muirhead in 1887.) ..."

 

[vanhees71]

Newton I and II are not limited to interactive forces without Newton III because Newton III and only Newton III excludes fictitious forces. Without this limitation to interactive forces there is no limitation to inertial frames.

Newton I is about particles moving without the action of any forces, Newton II defines forces as $\dot{\vec{p}}$ where $\vec{p}=m \vec{v}$, and Newton III is about two-body interactions. Indeed to state Newton I + II you don't need Newton III. That's all I'm saying the whole time!

The only thing, I'm now thinking about is, what about generic three-body forces (as occur in nuclear physics)? I guess these you can adequately only treat with analytical mechanics and symmetry principles to be consistent with Newtonian spacetime structure.

 

[vanhees71]

If you take the second law as the definition of force, it is defined as the rate of momentum change. The third law than says that for two interacting bodies, their rates of momentum change are always opposite but equal.

If our interaction is not instantaneous but mediated by a field, that's not true on an instant by instant basis anymore because the field carries non-zero momentum at intermediate times. I.e. the momentum gain of one of the bodies happens later than the momentum loss of the other.

Yes, and that's why fields are not belonging to Newtonian but relativistic physics ;-).

 

[vanhees71]

Griffiths also gives a general argument against the third law in relativistic theories in his "Introduction to Electrodynamics" (see chapter 12.2.4 "Relativistic dynamics"):
"Unlike the first two, Newton’s third law does not, in general, extend to the relativistic domain. Indeed, if the two objects in question are separated in space, the third law is incompatible with the relativity of simultaneity. For suppose the force of $A$ on $B$ at some instant $t$ is $F(t)$, and the force of $B$ on $A$ at the same instant is $−F(t)$; then the third law applies, in this reference frame. But a moving observer will report that these equal and opposite forces occurred at different times; in his system, therefore, the third law is violated."

Well, but in special relativity due to the homogeneity of space in any inertial frame still momentum conservation holds true and that's why the most natural description in relativistic dynamics are local field theories, and that in fact are the only hitherto successful formulation of relativistic dynamics, i.e., in some sense Newton III is still valid locally.

 

[DrStupid]

What if @vanhees71 replaced "non-interacting particles" by "force-free particles" in his post #57?

How do you know that the particle is force-free?

 

[DrStupid]

Chapter 1 of Lagrangian and Hamiltonian Mechanics by MG Calkin begins "Newton's first law deals with non-interacting bodies. It says that the velocity of an isolated body, one removed from the influence of other bodies, is constant. This law defines a set of preferred coordinate frames, inertial frames, as frames in which Newton's first law holds." https://www.worldscientific.com/doi/pdf/10.1142/9789810248154_0001

This is true because the force as used in Newton I is defined to be interactive by Newton III. Without Newton III there would be no justification for this statement.

N3 doesn't even apply to non-interacting bodies.

That depends on your definition of "apply". N3 says that non-interacting bodies are force-free. You are using this fact in your derivation of inertial frames from N1.

N1 is not a consequence of N3.

As nobody claimed something like that it seems we are talking cross-purposes. Maybe it helps when I explain it with an example:

Let's say I have a frame of reference where a single particle remains at rest no matter where I place it. That means according to

Newton I: As the particle remains ar rest, there is no force acting on it.
Newton II: As the acceleration is zero, the force acting on the particle is $F = m \cdot a = 0$

Now I switch to another frame of reference that is rotating around the origin with the angular velocity $\omega$. In this frame the particle is moving on circular paths around the rotational axis. That means according to

Newton I: As the particle doesn't remain at rest or uniform translation, there is a force acting on it.
Newton II: As the acceleration is $- \omega ^2 \cdot r$, the force acting on the particle is $F = m \cdot a = - m \cdot \omega ^2 \cdot r$

That's it. There is nothing in Newton I or II that tells me this is not allowed. That changes with

Newton III: As there is just a single particle, there is no interaction between particles and therefore no force.

The "forces" resulting from Newton I or II are violating Newton III. Or vice versa: The absense of forces resulting from Newton III violates Newton I and II. The laws of motion are not valid in the rotating frame. If inertial frames of references are defined by compliance with the laws of motion this means that it is not inertial.

The laws of motion define inertial frames of reference with (N1 or N2) and N3 but not with (N1 or N2) xor N3.

 

[DrStupid]

Interesting history of the concept of inertial frame: https://plato.stanford.edu/entries/spacetime-iframes/

"[...]Moreover, like Lange's, it leaves out a crucial condition for an inertial system as we understand it: all forces must belong to action-reaction pairs. [...] Therefore the definition needs to be completed by the stipulation that to every action there is an equal and opposite reaction. [...]

Thank you for this reference. This is exactly what I am talking about.

 

[Dale]

This is true because the force as used in Newton I is defined to be interactive by Newton III. Without Newton III there would be no justification for this statement

Sorry @DrStupid, I will stick with Dr Calkin here. His approach seems much clearer than yours.

You are using this fact in your derivation of inertial frames from N1.

I don’t derive definitions.

Newton I: As the particle doesn't remain at rest or uniform translation, there is a force acting on it.

Per the cited approach, since you have a non-interacting particle which is accelerating then your reference frame is non inertial. That seems the most direct approach to me.

Sure, you can go out of your way to use a poor formulation of Newton’s laws which requires you to use all of the laws for even the simplest cases, but you certainly don’t have to and I personally don’t choose to.

 

[vanhees71]

How do you know that the particle is force-free?

This is the crux ;-)). You have to establish an inertial reference frame first, according to the definition of Newton I. Then you assume that particles far away from any other particles are "force-free" and check whether all such particles move in rectilinear uniform motion relative to each other. You need one moving particle to establish a measure of time through the measure of distance and then at least two other particles moving in different directions to establish that you are in an inertial reference frame. Then you have the definition of force together with the definition of mass, which by assumption is a measure of "amount of matter".

 

[bhobba]

If our interaction is not instantaneous but mediated by a field, that's not true on an instant by instant basis anymore because the field carries non-zero momentum at intermediate times. I.e. the momentum gain of one of the bodies happens later than the momentum loss of the other.

Newtonian Gravity?

Thanks
Bill

 

[vanhees71]

Newtonian gravity is not a field theory but action at a distance, with the interaction potential
$$V(\vec{r}_1,\vec{r}_2)=-\frac{\gamma m_1 m_2}{|\vec{r}_1-\vec{r}_2|}.$$
Here the total momentum of a closed system of point particles is conserved, and no momentum is exchanged with any dynamical field. Newtonian Gravity is not a field theory in the narrower sense of a theory involving dynamical fields.