What Do Newton's Laws Say When Carefully Analysed

[DrStupid]

Newton I defines in an instrumental way what an inertial reference frame is and assumes that there exists one, and this establishes what Newton calls absolute space and absolute time, which are not in any way affected by anything happening by assumption.

That doesn't work without Newton III.

 

[Dale]

I can’t find it now, but I recall reading a very interesting approach to Newton’s three laws. It started with Newton’s first by defining a reference frame such that objects which were not interacting with any other object traveled in straight line at constant speed as inertial. Then it jumped to Newton’s third and when two objects interact only with each other they accelerate in opposite directions by an amount that are proportional with a constant of proportionality which is fixed for the two objects, and this constant of proportionality defines the ratio of their masses. Then they went to Newton’s 2nd law to define forces.

It was interesting, but I cannot find it now and I probably mis-remembered some crucial detail.

 

[vanhees71]

That doesn't work without Newton III.

What doesn't work out without Newton III? You can state Newton III only in connection with Newton I and II, as I tried to argue about in my previous posting.

 

[DrStupid]

It started with Newton’s first by defining a reference frame such that objects which were not interacting with any other object traveled in straight line at constant speed as inertial.

Same as above: Newton I doesn't define such a reference frame without the first part of Newton III ("forces act always pairwise"). As Newton deleted this part from the 3rd law [http://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/49] this interpretation of the 1st law doesn't seem to fit his intentions.

 

[DrStupid]

What doesn't work out without Newton III?

Defining inertial frames of references.

 

[Dale]

Same as above: Newton I doesn't define such a reference frame without the first part of Newton III ("forces act always pairwise"). As Newton deleted this part from the 3rd law [http://cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/49] this interpretation of the 1st law doesn't seem to fit his intentions.

The article I seem to remember was not interested in fitting his original intentions at all. It was about making everything rigorous. So in the laws they needed to define mass and force and inertial frames and how forces appear in equal and opposite pairs and add and maybe something else I am forgetting. It was definitely a more modern approach than historical.

I wish I could find it.

 

[Swamp Thing]

Is there a closed-form Cartesian equation for the 2-D curve that generates Norton's dome?

Some criticisms of Norton's Notion point to the singularity at the apex, i.e. absence of the second derivative. This leads to the question, if we attempt to build the dome, what kind of physical imperfections non-idealities would come into play to eliminate the paradox -- what would actually happen?

 

[DrStupid]

The article I seem to remember was not interested in fitting his original intentions at all.

That wouldn't be Newton's laws of motion anymore.

 

[Dr. Courtney]

The article I seem to remember was not interested in fitting his original intentions at all. It was about making everything rigorous. So in the laws they needed to define mass and force and inertial frames and how forces appear in equal and opposite pairs and add and maybe something else I am forgetting. It was definitely a more modern approach than historical.

I wish I could find it.

I remember something like that in Physics Today 20-25 years ago.

 

[Dale]

That wouldn't be Newton's laws of motion anymore.

This is science not history. The seminal authors get the first word about their theory, not the last.

 

[DrStupid]

This is science not history.

That includes that different laws must not be confused. Inventing new laws of motion is no problem. Euler did it for example. But he didn't call them Newton's laws of motion. They are known as Euler's laws of motion.

 

[vanhees71]

Defining inertial frames of references.

Why do you need Newton III for that? You just need non-interacting point particles moving relative to an inertial frame in uniform linear motion. You need one particle to define a measure of time in terms of a measure of distance (the latter given by Newton's tacit assumption that his absolute space is described as a 3D Euclidean affine manifold). Then all other non-interacting particles must move also in uniform linear motion given that measure of time, if your reference frame is an inertial frame.

So far, Newton did not need to quantify "interaction" or "force". This is done in Newton II, and then Newton III can be stated for interactions between particles.

You can also start from Newton I only, which defines the mathematical structure of Newtonian spacetime (a fibre-bundle construction) and then looking for dynamical laws that obey the 10D Lie-group symmetry of this spacetime model.

 

[DrStupid]

You just need non-interacting point particles moving relative to an inertial frame in uniform linear motion.

You need an inertial frame to define an inertial frame? That doesn't make much sense.

 

[vanhees71]

No you need non-interacting particles to define what an inertial frame is. I guess, my formulation above was a bit misleading ;-).

 

[DrStupid]

No you need non-interacting particles to define what an inertial frame is.

OK, let's say you have your non-interacting particles. How do you define an inertial frame with Newton I but without Newton III?

 

[Dale]

OK, let's say you have your non-interacting particles. How do you define an inertial frame with Newton I but without Newton III?

Newton’s 3rd law is irrelevant for non interacting particles.

 

[DrStupid]

Newton’s 3rd law is irrelevant for non interacting particles.

And that's irrelevant for my question. vanhees71 claimed that inertial frames can be defined with Newton I and without Newton III. Just let him explain how it works or do it yourself if you also think it is possible.

 

[Dale]

OK, let's say you have your non-interacting particles. How do you define an inertial frame with Newton I but without Newton III?

or do it yourself if you also think it is possible.

An inertial frame is a reference frame where all non interacting particles travel in straight lines at constant velocity.

 

[DrStupid]

An inertial frame is a reference frame where all non interacting particles travel in straight lines at constant velocity.

How does that follow from the first law?

 

[Dale]

How does that follow from the first law?

A non-interacting object has no external force. The first law says such objects travel in a straight line at constant speed.

 

[atyy]

No you need non-interacting particles to define what an inertial frame is. I guess, my formulation above was a bit misleading ;-).

I think what @DrStupid is saying is that N1 and N2 alone are consistent with a noninertial frame, provided we allow inertial forces (ie. acceleration is like gravity, the EP). However, N3 rules out inertial forces, which is why one needs N3 together with N2 (and N1 is a special case of N2) to define an inertial frame.

 

[Dale]

N1 and N2 alone are consistent with a noninertial frame, provided we allow inertial forces

Hmm, I am not sure I agree. N2 certainly is consistent with a non inertial frame, but in a non inertial frame all objects are subject to the inertial forces, so you can never get a force-free object on which to use N1.

N1 stipulates no force, which is more restrictive than no net force. That stipulation I think eliminates non inertial frames, which is why it can be taken as a definition of an inertial frame.

However, for clarity I prefer an explicit reference to inertial frames as above.

 

[atyy]

Hmm, I am not sure I agree. N2 certainly is consistent with a non inertial frame, but in a non inertial frame all objects are subject to the inertial forces, so you can never get a force-free object on which to use N1.

Yes, but then it means that to define an inertial frame, one needs force-free objects. However, force-free objects may not exist.

 

[olgerm]

let's say you have your non-interacting particles. How do you define an inertial frame with Newton I but without Newton III?

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.

Since no force can act upon non-interacting particles:
A frame is inertial if in it all non-interacting particles are at rest or moving with constant velocity.

 

[bhobba]

And that's irrelevant for my question. vanhees71 claimed that inertial frames can be defined with Newton I and without Newton III. Just let him explain how it works or do it yourself if you also think it is possible.

See Landau - Mechanics - its clear and all based on symmetry principles - nothing else required. It's basically how I explained it, which is a special case of the POR ie being careful with what exactly an inertial frame is before stating the POR. The bit about free particles moving with constant velocity follows from the PLA - my explanation is a bit hand-wavy. The only thing I will mention is I am not 100% happy with his derivation of the Lagrangian - I prefer it as the classical limit of the relativistic Lagrangian - but I think that's just a personal preference.

The key here is to define an inertial frame carefully then derive free, non interacting, whatever you want to call it, particles move at constant velocity

Thanks
Bill

 

[bhobba]

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.

Since no force can act upon non-interacting particles:
A frame is inertial if in it all non-interacting are at rest or moving with constant velocity.

Hmmm. People reading that will know what you are getting at, but the definition using symmetry allows its symmetry properties, necessary to apply Noether, to be explicit. It's harder to get those symmetry properties from the above - in fact I do not know how its done because it's statement involves the laws of physics.

Thanks
Bill

 

[vanhees71]

OK, let's say you have your non-interacting particles. How do you define an inertial frame with Newton I but without Newton III?

I already tried to make that clear, but here again:

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, i.e., you have to think about whether there's either an interaction you have overlooked, e.g., if you take a rest frame relative to Earth (e.g., your lab) you have to take into account the gravitational interaction of the particle with the Earth.

Now suppose you have a particle running in a straight line, given the assumption that space is described by a Euclidean affine manifold and thus you can measure distances, you have a definition of time intervals through the time it takes the particle to run a certain distance.

Now you can check whether all other non-interacting particles also run in uniform linear motion, and then you are sure to have defined an inertial reference frame (at least within the accuracy you are able to do all the measurements described above). Nowhere did I need Newton II no Newton III. Newton II is needed to define forces, though this is not independent of the notion of mass, and that's why Newton II introduces both force and mass and then Newton III is also well-defined through the notion of force.

 

[atyy]

See Landau - Mechanics - its clear and all based on symmetry principles - nothing else required. It's basically how I explained it, which is a special case of the POR ie being careful with what exactly an inertial frame is before stating the POR. The bit about free particles moving with constant velocity follows from the PLA - my explanation is a bit hand-wavy. The only thing I will mention is I am not 100% happy with his derivation of the Lagrangian - I prefer it as the classical limit of the relativistic Lagrangian - but I think that's just a personal preference.

The key here is to define an inertial frame carefully then derive free, non interacting, whatever you want to call it, particles move at constant velocity

Here what you are saying is not so different from what @DrStupid is saying. You are saying that first you need a notion of the symmetry of the laws, which is heuristically similar to what @DrStupid is saying that N3 is needed. The reason is that N3 is momentum conservation, which is equivalent to a symmetry via Noether's theorem.

 

[vanhees71]

Sure, you can also start from Newtonian space-time structure, defined by the 10D Galilei group as a symmetry (this is very much the approach to geometry as advertised in Klein's "Erlanger Programm"). You can take this as the abstract mathematical version for Newton 1. Then Newton III follows from the homogeneity of space, i.e., momentum conservation in the special case of pair-wise interactions only.

 

[Dale]

Yes, but then it means that to define an inertial frame, one needs force-free objects. However, force-free objects may not exist.

Agreed. The force-free objects are great for definitions, but not so great for practical implementation.