A paradox inside Newtonian world

[Tomaz Kristan]

Newton's laws don't say anything about the existence of point particles or when their use is valid.

They don't say what they are, they don't say what they aren't, just how they behave.

One could certainly makes up a construction like this one. It's legal, like many others which do not make any troubles, however.

 

[vanesch]

I`ll just repeat what I said before. It's a nice curiosity but doesn't contain any physics.

Sure, I agree with that. But there's nothing wrong with pointing out a curiosity !

Continuum mechanics is just an approximation, a method for dealing with a large number of particles. Please don't mix up the mathematics with the physics.

Well, given that matter is made of atoms and so on, continuum mechanics is physically approximative. But I saw that rather as a boundary to its domain of applicability, and not as a kind of "approximation": I see continuum mechanics as a field theory, where the field is the matter density (as a genuine continuum without any "particle", which would then be a Dirac pulse in the matter density field), and the stress tensor field, in the same way as you can see the electric field or the magnetic field.

So, continuum mechanics, as an abstract field theory, might be self-consistent or not. Particle mechanics is a special case of it, where we have dirac-delta functions as the matter density.
The question is: what kind of functions is allowed for, so that the theory is well-defined.

 

[Tomaz Kristan]

We could have a cone. Perfect cone of a small mass and then we could insert those balls inside the cone. The infinite number of them, ever smaller, ever denser, to the top.

We could stuff a pyramid with those balls, also.

It would hover in the mid air above the planet.

That would be a curiosity!

[Inside a Newtonian world, of course.]

 

[vanesch]

We could have a cone. Perfect cone of a small mass and then we could insert those balls inside the cone. The infinite number of them, ever smaller, ever denser, to the top.

We could stuff a pyramid with those balls, also.

It would hover in the mid air above the planet.

That would be a curiosity!

[Inside a Newtonian world, of course.]

I like that vision

But the day that someone sums the forces differently, it might crash down and expose the tomb of the Pharaoh

 

[Hurkyl]

One could certainly makes up a construction like this one. It's legal

That's the whole problem there: why do you think it's legal? You've made two key assumptions on particle mechanics:

(1) Any arrangement of countably many particles is valid.
(2) Conservation of momentum holds for any particle configuration.

I've tried to prompt you on those, but you responded like I was insane! Why would anyone ask a stupid question like that??

But that's the most important question here; the thrust of your work is that you've shown these two assumptions to be inconsistent with each other. And thus, a valid formulation of Newtonian mechanics cannot include both.

But it seems you're not interested in using this to pursue what a valid formulation of Newton mechanics might look like; you seem more interested in preserving this invalid formulation so you can say "Hey look, a paradox!".

 

[vanesch]

(1) Any arrangement of countably many particles is valid.
(2) Conservation of momentum holds for any particle configuration.

Well, the problem as I see it, is that from the moment that you allow for continuum mechanics as a field theory (where a finite amount of mass is distributed according to just any "reasonable" distribution), from the moment that you impose certain conditions on these distributions (which conditions ? I would vote for non-divergent mass density), you have to make sure that they cannot be the result of the dynamical evolution of an "allowed-for" distribution (in other words, that your conditions are invariant under dynamical evolution).

The "conservation of momentum" is just another way of summing the same set of forces (which constitute a conditionally convergent series). You can sum them pair-wise (action-reaction), and then you find obviously zero.

But that's the most important question here; the thrust of your work is that you've shown these two assumptions to be inconsistent with each other. And thus, a valid formulation of Newtonian mechanics cannot include both.

Yes, and that's the interesting curiosity, isn't it ?

 

[Tomaz Kristan]

Our disagreement is not big, at all.

The only thing we three don't agree is, that I see the Lagrangian or Hamiltonian mechanics as Newtonism with both those assertions assumed.

You two don't.

Not a big difference.

 

[Hurkyl]

You two don't.

Not a big difference.

Really, it is. Inconsistency is a big deal. Given your assumptions, you can prove Jupiter orbits the sun in a hyperbolic orbit, electromagnetism doesn't exist, and that there's a perpetual motion machine in my back yard. That's why paradoxes are a bad thing to have.

 

[SF]

I recently found this on NewScientist: http://www.newscientist.com/article.ns?id=mg19225802.000&feedId=fundamentals_rss20

It asks: You'd think we'd know how heat flows and pendulums swing - so how come we've got it all wrong?

I don't have a NS account, what do they mean? Is this a part of classical mechanics/thermodynamics we didn't understand properly?

 

[Alkatran]

Let me summarize the conversation up to now and into the future.

"It's wrong"
"No it isn't"
repeat

 

[vanesch]

Let me summarize the conversation up to now and into the future.

"It's wrong"
"No it isn't"
repeat

Tomaz' argument isn't wrong: the example he cites can be thought of as existing in a certain kind of Newtonian world and indeed, the total force on the system is given by a conditionally convergent series (in other words, its value is undertermined, because depending on the way we order the sum, which isn't specified by any of Newton's postulates). As such, I find this an interesting example.

Now, as pointed out by others, there is few chance that this kind of situation appears in any practical application of Newtonian mechanics, so there's no problem for a practical mind.

As still others pointed out, Newton's postulates do not constitute a formal system. There's a simple way to eliminate Tomaz' example: require a finite number of point particles. My objection to that is that informally, we also use Newton's system in continuum mechanics where this requirement is not respected.

As such, in order to build a formal system which can include the practical application of continuum mechanics (which is really used), one should find exactly those conditions on the acceptable situations which avoid Tomaz' example. The only thing I could point out is that the mass volume density diverges in his example.

So let's say that Tomaz' example is interesting because it puts a limit to the kind of formal system one could build around Newton's postulates (a bit in the same way as Russell's paradox puts some constraints on how one can formalize the informal set theory by Cantor).

 

[vanesch]

Really, it is. Inconsistency is a big deal. Given your assumptions, you can prove Jupiter orbits the sun in a hyperbolic orbit, electromagnetism doesn't exist, and that there's a perpetual motion machine in my back yard. That's why paradoxes are a bad thing to have.

Ok, but according to you, what is the origin of the paradox ?
(the paradox being that applying Newton's postulates to the situation as described by Tomaz, they only give us a means to construct a conditionally convergent series for the total force on the system)

Note that there doesn't need to be an assumption of momentum conservation: it simply follows from another way of summing the same set of forces - which shouldn't surprise us, given that it is conditionally convergent. The real paradox resides in the appearance in the first place of a conditionally convergent series without a clear prescription of the order in which we have to sum the terms.

And how do you propose we eliminate it ?

 

[Hurkyl]

Ok, but according to you, what is the origin of the paradox ?

As I said in #215, the paradox arises because we assume both:

(1) Any arrangement of countably many particles is valid,
(2) Conservation of momentum holds for any particle configuration.

And how do you propose we eliminate it ?

I haven't had the time to mull it over yet.

 

[Tomaz Kristan]

a bit in the same way as Russell's paradox puts some constraints on how one can formalize the informal set theory by Cantor

I like this one. I agree. ZF + Newton laws is an inconsistent system.

 

[Alkatran]

Tomaz' argument isn't wrong: the example he cites can be thought of as existing in a certain kind of Newtonian world and indeed, the total force on the system is given by a conditionally convergent series (in other words, its value is undertermined, because depending on the way we order the sum, which isn't specified by any of Newton's postulates). As such, I find this an interesting example.

..

I do believe this post replying to my post qualifies as a self-fulfilled prophecy.

 

[vanesch]

I do believe this post replying to my post qualifies as a self-fulfilled prophecy.

Who is claiming that Tomaz' example is "wrong" ?
In the beginning of this thread, yes, there were remarks of the kind "you must have made a computation error or something".

Then it was realized that the setup (the mass distribution) gave rise to a set of forces which was conditionally convergent:
if you sum them pair-wise (that is: "adding balls"), the sum cancels ;
if you sum them "first all forces over ball 1, then over ball 2"... you get a finite or infinite force to one side.

This behaviour of a conditionally convergent series is known: by re-arranging the terms, we can obtain any result we want. But it is not because this is known, that the problem goes away! So when a conditionally convergent series appears, one would need also a rule that tells you in what order you should sum it, if you need a value. I provided another example (the inflating sphere) where the trick that could work in Tomaz' case, doesn't work for my example.

So the claim that the given mass distribution (which can be correctly described as a distribution over an Euclidean space, except that it diverges around 0) gives rise to an indeterminacy as to what force acts as a whole on the distribution, because if you apply Newton's postulates, it spits out a conditionally convergent series, is a correct claim, no ?

I think the genuine thing to ask for is: what extra condition does a mass distribution need to have such that:
1) no conditionally convergent series of forces is generated by it
2) the condition is "stable under time evolution"
3) the condition is friendly enough to encompass all practical applications of Newtonian mechanics (such as continuum mechanics).

 

[Alkatran]

Who is claiming that Tomaz' example is "wrong" ?
In the beginning of this thread, yes, there were remarks of the kind "you must have made a computation error or something".

I was commenting on the fact that you took my comment as meaning "it's wrong" and then said the equivalent of "no it isn't".

 

[Tomaz Kristan]

Constructions like mine, are not possible in the real world. Nothing paradoxical is possible in the real world, of course.

It is interesting, how physics forbids arbitrary dense particles by limiting the black hole radius. You can't compress one even more, than it already is. It's a clear limit here.

Hadn't been so, this construction might be doable and we would have a real life problem.

Discretization of mater and space - what Ernest Mach so opposed - may be a pure logical necessity!

 

[reilly]

Look, if A attracts B with force F, then B attracts A with an equal and opposite force. Period. Thus, there's no way the CM can move, even though the individual masses can move, cf planetary orbits. In addition there's Thomson's Thrm, which says the system described here is in unstable equilibrium. So, what is likely to happen is that every mass will "fall' toward the CM; what happens when they all get there is another matter. Note that if you include E&M radiation, which will be present for most matter, the CM then involves not only the masses, but the radiation as well.

This is a good AP physics exam question. The CM does not move: repeat, the CM does not move. If it did, then airplanes probably would not work, the moon might fall into the Earth -- at any moment, certainly rockets would not work. In short, the world would be a very different, terrifying willy-nilly world.

A relatively close analogy is the physics of infinite chains of springs undergoing small oscillations. Check it out for a dose of reality, and correct physics.

Regards,
Reilly Atkinson

(Note --
With all due respect, I will admit to not reading most of the posts. That there are more than 4 or 5 posts totally confounds me. The paradox is due to substantial errors in thinking about Newtonian physics; there is no paradox, not even close. )

 

[vanesch]

Look, if A attracts B with force F, then B attracts A with an equal and opposite force.

Well, usually you're right, but I think you went too fast on this one. Work out the forces (in the beginning of this thread, I provided some mathematica notebooks on it): indeed, if you sum them the way you do, then indeed the CM doesn't move.
However, if you first sum all the forces on one ball, and then all the forces on the second ball etc... and THEN sum all these total forces on all these balls, you get a non-zero result.

How can this be ? They are pair-wise zero !

The answer is that they make up a conditionally-convergent series and that the total sum depends on what order we are summing them !

Now, before you go through the entire discussion again, we all agree that this is not a real-world example, that this is not a practical problem, that this will never occur etc... However, as a FORMAL example in a purely theoretical setup, this is entirely possible.

As discovered a few times during this thread, yes, the mathematical reason is that the total set of forces forms a conditionally convergent series (where addition is not commutative anymore, if you like). This is the culprit.

So, what is likely to happen is that every mass will "fall' toward the CM; what happens when they all get there is another matter.

Well, be my guest, and calculate the acceleration of each mass... for that, you need to sum all the forces due to all the others, right ?
Well, it turns out that for ALL the masses, the sign of this force is the same...
(look at my mathematica notebooks if you will).

This is a good AP physics exam question. The CM does not move: repeat, the CM does not move. If it did, then airplanes probably would not work, the moon might fall into the Earth -- at any moment, certainly rockets would not work. In short, the world would be a very different, terrifying willy-nilly world.

This is what some people missed in this thread: this is not a problem in our universe: it is a problem in a Newtonian toy universe where Newton's laws strictly hold and only gravity works as an interaction.

With all due respect, I will admit to not reading most of the posts. That there are more than 4 or 5 posts totally confounds me. The paradox is due to substantial errors in thinking about Newtonian physics; there is no paradox, not even close. )

This is also how I got into this thread: I fired up mathematica to show Tomaz quickly wrong... to discover that he was right.

Again, it has nothing to do with our world, it doesn't imply anything for any practical application of Newtonian physics etc... But it is a problem in a Newtonian universe if you are too relaxed on what kinds of initial states you can allow for.

 

[reilly]

vanesch -- Yup. Conditionally convergent -- good eye, as we say in baseball. Missed that one. Still ... It seems reasonable that an "unphysical system" will cause difficulties.
Regards,
Reilly

 

[Tomaz Kristan]

The paradox is either in:

- Zermelo Fraenkel Set Theory

- Newton's laws

- a theorem derived from Newton's laws, contradicts a theorem from Zermelo Fraenkel Set Theory. They oppose each other.

My guess is ... oh, never mind! What's yours?

 

[Hurkyl]

The paradox is either in:

- Zermelo Fraenkel Set Theory

- Newton's laws

- a theorem derived from Newton's laws, contradicts a theorem from Zermelo Fraenkel Set Theory. They oppose each other.

My guess is ... oh, never mind! What's yours?

My guess is one of the following:
(1) Given your definition of matter distribution, and your interpretation of Newton's laws, conservation of momentum is not a theorem.

(2) Given your definition of matter distribution, and your interpretation of Newton's laws, conservation of momentum is a theorem, but this matter distribution does not satisfy the hypotheses of that theorem.

(3) Your definition of matter distribution and your interpretation of Newton's laws are shown by this example to be self-contradictory, which says nothing about how someone else might formulate classical mechanics.

 

[vanesch]

My guess is one of the following:
(1) Given your definition of matter distribution, and your interpretation of Newton's laws, conservation of momentum is not a theorem.

(2) Given your definition of matter distribution, and your interpretation of Newton's laws, conservation of momentum is a theorem, but this matter distribution does not satisfy the hypotheses of that theorem.

(3) Your definition of matter distribution and your interpretation of Newton's laws are shown by this example to be self-contradictory, which says nothing about how someone else might formulate classical mechanics.

No, conservation of momentum is not a theorem anymore given these distributions of mass (and their gravitational interactions). Because the proof of the conservation of momentum only proves that the forces are pair-wise zero, which doesn't prove that their overall sum is zero when the total series of forces is a conditionally convergent series, because in that case, it depends in what order one sums them, which is left, by Newton's laws, up to the good judgement of the user. It is simply this, which is the cullprit: the fact that certain distributions of mass, together with the usual gravitational force law, gives rise to a series of forces which constitutes a conditionally convergent series, without any further specification of in what order one should sum them - hence giving rise to an ill-defined value for the "total force" which one needs in the equations of motion.

So the problem is simply that for certain mass distributions, the laws of Newton don't prescribe in a unique way what are all the total forces, and hence one cannot write down in a unique way, the equations of motion. By making (arbitary) choices, one arrives, depending on the choice, at different equations of motion, which is what the paradox is all about.

 

[reilly]

This problem has been driving me nuts. What I'm about to say, has already been basically said by vanesch & others. But I wanted to get an approach that made sense to me. So...

For any finite system of masses -- finite number, finite masses -- embedded on the 0-1 interval we know that both energy and momentum are conserved if the masses interact through central potentials. Under these circumstances, the CM, if initially at reast, will not move -- even with collisions inside ... given appropriate assumptions about symmetries and mass-mass contact forces.
This situation will hold for infinite systems, provided all necessary sums or integrals converge. If z=1/10, and r is the coordinate of an arbitrary point on the 0-1 interval, then the potential of the nth particle, apart from signs and constants, is V(n) = (5z)**n/{ |r - z**n|}. For any valid non zero r -- the sum over n of V(n) will be finite. but for r = 0, the ballgame is over, and the sum over n of V(n, r=0) is badly divergent. ( For those interested in such things, the singularity at r=0 is, some kind of branch point, as n -> infinity, the number of singularities becomes denser and denser as r - > 0. So the singularity at r=0 is not an isolated singularity. For clarity, we are talking about the singularity in r of the sum of V(n) over n.)

Any way, if we chose 0-1 as open at r =0, then everything converges, and the problem is mathematically well set, but not so physically.
If we choose the 0-1 interval closed at 0, then the problem is not mathematically well-set because of a diverging potential at r=0.

Note that line-charge potentials do not act well at the endpoints, typically because of logarithmic singularities -- branch points.Strictly speaking, Newton's laws are valid for finite systems, and infinite systems, which are convergent -- effectively, the mass points and potential values should form a compact set. Without guaranteed convergence, it's a crap shoot, and anyone knowledgeable about QFT will surely concur.

The paradox is trying to fit a physically impossible system into a physically correct description. Newton's laws are alive and well, and are valid for real physical systems. Be very wary of Greeks, or anyone else, bearing infinite gifts wrapped in paradox.
Regards,
Reilly Atkinson

 

[vanesch]

Strictly speaking, Newton's laws are valid for finite systems, and infinite systems, which are convergent -- effectively, the mass points and potential values should form a compact set. Without guaranteed convergence, it's a crap shoot, and anyone knowledgeable about QFT will surely concur.

Yes, however, the question that arrises is then: can we guarantee that a system whose initial conditions satisfy those requirements, will keep satisfying them through time evolution ?

 

[Tomaz Kristan]

Well, I knew all the time, that only something bad will come out of the Infinity Hotel.

Mathematicians were all too careless about that, at least since Cantor introduced that annoying thing.

Then I've said to myself, let translate the Infinity Hotel story, where you can transfer money down from room number N to room number round(N/100000). Everybody has millions of dollars, where just a few bucks where before ... and a lot of similar crap happens at those hotels.

... let translate this to a nice physical object, with a finite mass even, inside the abstract Newtonian world! Should of work!

So, instead of a hotel room for every number, we have a mass point (or ball even) for every number. Instead of a money transfer we have the gravity, for example.

The mess is more obvious, however. More difficult to swallow, than those Cantor's exercises, which only the intuitionists and finitists find ... improbable.

 

[Tomaz Kristan]

can we guarantee that a system whose initial conditions satisfy those requirements, will keep satisfying them through time evolution ?

No, I can easily imagine, how to transform something quite innocent, into this construct.

 

[Hurkyl]

Well, I knew all the time, that only something bad will come out of the Infinity Hotel.

Mathematicians were all too careless about that, at least since Cantor introduced that annoying thing.

Who's being careless here? Is it the people who study the subject with care and rigor? Or is it the guy complaining that it disagrees with his intuition?

 

[Tomaz Kristan]

Who's being careless here? Is it the people who study the subject with care and rigor?

What's the use of a rigor, if the axioms contradicts each other?

I am not saying here, that I see a direct evidence of that in ZF Set theory. I am saying, I see it in ZF+NL.