A paradox inside Newtonian world

[Tomaz Kristan]

When using informal means, if you derive a paradox, you have not proven your axioms are self-contradictory.

Newton Laws ARE an example of a formal system. Not consistent with the math, inside which are presented, as we saw.

 

[Tomaz Kristan]

The problem here is not a physics problem, it is a mathematics problem dressed up to look like a physics problem.

Physics can't have a practical problem. It is, how it is. A physics theory can be flawed.

The pure Newtonism is.

 

[vanesch]

In particular, you may have applied your heuristics outside of their domain of applicability. Searching for paradoxes like this is, in fact, a typical way of discovering the limits of an informally expressed idea.

Yes, that's why I found this example interesting, without being worrysome. Clearly, Newton's laws, with Newton's gravity, cannot be applied to just any mass distribution ; this is what this example demonstrates. However, the question is: how "liberal" can one be with the mass distributions before problems set in...

 

[Tomaz Kristan]

Some people already worry about the so called Seeliger's paradox. An infinite Euclidean universe uniformly filled with mass, leads to the infinite gravity force at every point ... and so forth.

http://www.google.com/search?hl=en&lr=&safe=active&q=seeliger's+paradox&btnG=Search

 

[reilly]

Physics can't have a practical problem. It is, how it is. A physics theory can be flawed.

The pure Newtonism is.

A bit hard to follow. What does your first sentence mean? Strikes me that physics is full of practical problems -- designing and building particle accelerators or electron microscopes, ballistics, ...

Not only can physics theories be flawed, but physicists themselves as well.

But, in spite of any formal problems, Newton and Maxwell seem to work just fine in our everyday world -- airplanes, computers and the internet, radio and TV, highway design, ... That's good enough for most physicists. What's the problem? Does it have bearing on how we use physics in our everyday world, or in getting tenure?

Curious minds want to know?

Regards,
Reilly Atkinson

 

[Tomaz Kristan]

A bit hard to follow. What does your first sentence mean?

Physics in reality, can't harbor a paradox inside. A theory can and is then wrong in a formal sense. And not in accordance with the real world.

 

[Lonewolf]

Physics in reality, can't harbor a paradox inside. A theory can and is then wrong in a formal sense. And not in accordance with the real world.

You do realize that your set up of masses couldn't exist in the "real world", don't you?

 

[Tomaz Kristan]

You do realize that your set up of masses couldn't exist in the "real world", don't you?

Yes, I do. I've mentioned this, too.

 

[reilly]

Physics can't have a practical problem.


The pure Newtonism is.

As the great Count Basie once said: One more once. I don't get it, what kind of practical problems are you talking about? Why can't physics have them?

Please, give us an answer.

Regards,
Reilly Atkinson

 

[Tomaz Kristan]

I don't get it, what kind of practical problems are you talking about? Why can't physics have them?

A problem is always in inadequate theory and only there. I realize, you can't have this construct in practice, something is wrong only inside the Newtonian abstraction of the real world. NOT in the real world.

Okay now?

 

[ObsessiveMathsFreak]

A problem is always in inadequate theory and only there. I realize, you can't have this construct in practice, something is wrong only inside the Newtonian abstraction of the real world. NOT in the real world.

Okay now?

Prove it. You still haven't. All we're getting is handwaving. You haven't even put down one single equation in over 250 posts.

Is it so much to ask that you come up with the equations of motion for your system. Something similar to post 136? Because all we're getting, and all we have been getting since about then from your posts is denial. Please present a valid list of equations(not prose).

 

[Tomaz Kristan]

OMF,

Why should I?

You understand, or you don't. You are on denial, or you aren't.

What would change, a bunch of fancy expressions?

 

[reilly]

A problem is always in inadequate theory and only there. I realize, you can't have this construct in practice, something is wrong only inside the Newtonian abstraction of the real world. NOT in the real world.

Okay now?

No. Why? Bye.
Regards,
Reilly Atkinson

 

[Tomaz Kristan]

No. Why? Bye.

Is this is the way, how the problem will go away?

 

[reilly]

What problem? Oops, my big mouth took over.

If it is such a big and profound problem. write a paper. Or, with all due respect, take yourself less seriously.

 

[Tomaz Kristan]

So you say, there is no problem here?

Or you say there is?

What's your position, really?

 

[George Jones]

I don't know how relevant this is, but the astrophysicist John Barrow, in his book Impossibility: The Limits of Science and the Science of Limits, writew

Time and time again, the development of our most powerful theories has followed this path: ... then something unexpected happens. The theory predicts that it cannot predict: it tells us that there are things that it cannot tell us. Curiously, it is only our most powerful theories that seem to possesses this self-critical feature. ... I believe that we can expect to find more of these deep results which limit what can be known.

He has singularities in general relativity, probabilities in quantum theory, incompleteness in formal systems, and unknown aspects of new theories in mind when he makes these comments.

 

[reilly]

Although I've spelled out my take on the matter above, I'll try once again. The primary issue is convergence of infinite series. As vanesch pointed out, the problem is one of conditional convergence -- Is it a paradox for a series to have different limits? -- Hurkyl has also been right on target. My take involves the difference between open and closed intervals, a distinction not commonly used in physics -- can't find them in nature. And, so far, no one has found a system with an infinite number of particles.

Consider: the boy is a dog. There are at least two meanings to this sentence, a problem characteristic of many languages, including math.

Paradoxes, no doubt, abound in physics. For classical mechanics more fruitful ground might be found in mechanics of continuous media, or in the chaotic difficulties in the mechanics of more than two bodies. Yours is not a paradox.

If you want to convince us, then get to the real issue of convergence -- the normal equal and opposite property of central forces does not hold for the closed interval. Thus the closed interval version has nothing to do with physics.

Regards,
Reilly Atkinson

 

[Tomaz Kristan]

And, so far, no one has found a system with an infinite number of particles.

Of course not. But inside the Newton's abstract world, you have them as many as you wish. Infinity of points is 100% legal.

Paradoxes, no doubt, abound in physics.

They are forbidden. As everywhere. Something may have a "paradox" in name, but no real paradox is allowed.

Yours is not a paradox.

Says you. But have no idea, how the explain it.

If you want to convince us

Well, I don't care for everybody, if she is convinced or nor. I care for a possible hidden error in my line of thinking. No one has been able to show me one. This is a valuable information for me.

the normal equal and opposite property of central forces does not hold for the closed interval.

We have an half open interval here. But this has no role at all.

Thus the closed interval version has nothing to do with physics.

Where did you get this one?

 

[Hurkyl]

Infinity of points is 100% legal.

Prove it.

(and not just that infinitely many points is legal, but any configuration of infinitely many points is legal)

And prove that the center of mass theorem holds for your configuration of points.

 

[Tomaz Kristan]

Hurkyl,

There are NO exceptions. Every finite mass configuration is legal, no matter how bizarre it may looks to someone.

And every finite mass has it's center of gravity.

You think not?

 

[ObsessiveMathsFreak]

There are NO exceptions. Every finite mass configuration is legal, no matter how bizarre it may looks to someone.

And every finite mass has it's center of gravity.

You think not?

You dodged the question. You were asked to prove that an "infinte" configuration of masses is legal. You've only referred to a finite configuration of masses. That doesn't of itself show anything about an infite configuration of masses.

You still haven't proven that an infinite number of particles is a legal configuration. And moreover, given that it is, you still haven't proven that the center of mass calculation is valid either, and even then you still haven't given the motion of the center of mass under the gravity forces you describe.

 

[Tomaz Kristan]

You still haven't proven that an infinite number of particles is a legal configuration.

One ball is legal, isn't it? Divisible more than a finite number of times. Inside Euclidean space, where Mr. Newton operates.

QED.

 

[vanesch]

Of course not. But inside the Newton's abstract world, you have them as many as you wish. Infinity of points is 100% legal.

It is not the infinite number of points in itself that is a problem. It is the fact that your mass density diverges which is the cullprit. And not by itself, but in combination with a force law such as the gravitational law of Newton.
Now, before you say that a mass point has infinite density (which is true), it can often be replaced with a sphere of same mass and of finite density. This can be done each time when other mass points do not come nearer than a distance d in the problem.

The problem with infinite mass density is that the gravitational potential energy can diverge, from which moment onward anything goes. As such, a force law in 1/r^2 is NOT to be had in such a system. For instance, your system wouldn't (I think) give any problem if the interaction law were something like Hooke's law.

So, with a given force law (in casu 1/r^2), certain mass distributions are simply forbidden. In the same way, as with Hooke's law, mass points at divergent distances would be forbidden.

 

[Tomaz Kristan]

In the same way, as with Hooke's law, mass points at divergent distances would be forbidden.

Doe to the fact, the sum of divergent distances is not a finite distance. Can't fit into Euclidean space.

OTOH, 1/r^4 force between points is also legal. Since not illegal.

certain mass distributions are simply forbidden

Where and how and by whom or what?

 

[reilly]

Note Godel says that paradoxes cannot be avoided.

Physicists are a peculiar, pragmatic bunch when it comes to most anything.
There many things unexplained, say in functional analysis or set theory, but we simple-minded physicists don't worry much about such things, and, to a great extent rely on intuition -- Dirac's great intuition lead to the later theory of distributions. Dirac was right, but it took a lot of brilliant mathematicians to make a rigorous case for distributions. Renormalization -- totally intuitive, groping in the dark, but physicists muddle on, although progress is slow.

If you can show how your alleged paradox causes real practical problems in physics, in a very specific and clear fashion, then we'll listen carefully. And when I say physics, I mean physics as practiced by physicists, not mathematicians.

And, again, does a conditionally convergent series present a paradox? Do let us know, please.

Regards,
Reilly Atkinson

 

[Hurkyl]

Hurkyl,

There are NO exceptions. Every finite mass configuration is legal, no matter how bizarre it may looks to someone.

And every finite mass has it's center of gravity.

You claim a lot of things. Prove them.


(Incidentally, there are arrangements of countably many point particles whose total mass is finite, but don't have a center of gravity)

Edit: maybe I should give an example. If we permit the i-th particle (for any natural number i) to have mass 1/(2^i), and to be located at x-coordinate 2^i, then this configuration has total mass 1, but no center of mass.

 

[StatusX]

I haven't gone through this entire thread, but it's an interesting problem raised at the beginning. The OP seems to be claiming that any finite distribution of mass is acceptable since this just is a mathematical theory. If that's the case, to be rigorous you need to formalize the theory.

The easiest way to do this is to allow only point masses, and any number of them, just as long as the total mass is finite. It's easy to show the proof of Newton's 3rd law does not carry over to such a model. In that sense, the paradox is resolved (again, I haven't read the whole thread, so maybe this was already covered). One way to ammend this model is to require that particles not get arbitrarily close to one another, which is pretty reasonable on physical grounds. Another is to require only finitely many particles (note this is a strictly stronger condition than the previous one), which is also reasonable, albeit a bit more limiting.

If you were thinking of a different model, please be clear about what it is.

 

[DLinkage]

The sun is on the left, the Earth is on the right. Both are initially at rest.

The Earth moves to the right meaning the CM accelerates to the right...


Completely wrong. But the product of the Earth's mass and its velocity in one direction are equal the product of the sun's mass and its velocity in the other direction. Since the sun is so much mroe massive than the Earth it is hard to observe this.

This happens all the time in our solar system. The 9 planets probably create zero net force on the sun once every 10 trillion years (if the sun could even last that long). The sun is in its own orbit about the center of mass of our solar system. The radius of this orbit is tiny compared to the planets' orbits. But it does move,just like you chain of particles all will. In fact if you assume a perfectly elastic collision in one dimension you will find the particles will bounce off each other and oscillate. It would be quite the dynamics problem to figure out the time histories of each particle since they will all be hitting each other at different times before bounding back. I do predict though that the smallest ones would have the largest amplitude, although I doubt any notion could be made about frequencies since the modes will be the superposition of an infinite number of complex conjugate eignevalues

 

[Tomaz Kristan]

It is nothing remained to be proved here.

You prove to me, that there is ONE finite mass form, with no mass center!

You prove to me, that some finite "mass sculptures" are forbidden inside the Newton abstract world!

You prove to me, than ONE such exception was considered anywhere, anytime!

There is NOTHING like that at all. Everything should work for EVERY finite mass, distributed anyway you want inside a finite amount of space. No doubt about that.

Or please, show me the law, which forbids some constructions, like mine.

Yes, some limitations should be installed in the future. But currently, there are none. None such has been considered, since the Newton's time to the present.

Or please, name one. Just one.

 

[ObsessiveMathsFreak]

One ball is legal, isn't it? Divisible more than a finite number of times. Inside Euclidean space, where Mr. Newton operates.

QED.

QE what? What was that supposed to mean? "Divisible more than a finite number of times." That's not even wrong. You dodged the question again with nonsense.

Forget the validity of an infinite number of particles. How are you obtaining the convergence of the infinite series. Would you mind putting down even one solitary equation in support of your argument.

 

[vanesch]

Doe to the fact, the sum of divergent distances is not a finite distance. Can't fit into Euclidean space.

No, take the example of the point distribution by Hurkyl: mass points of mass 2^(-i) at position x = 2^i. For each i, this mass point exists in Euclidean space. Now consider Hooke's law between them. I didn't do the math, but I'm pretty sure we arrive at a similar conclusion as yours.

 

[Tomaz Kristan]

No, take the example of the point distribution by Hurkyl: mass points of mass 2^(-i) at position x = 2^i

Why should I?

For each i, this mass point exists in Euclidean space. Now consider Hooke's law between them. I didn't do the math, but I'm pretty sure we arrive at a similar conclusion as yours.

Even worse then. If it can go any worse than that, where a paradox has already occurred.

Its potential modification - doesn't solve a thing!

 

[Tomaz Kristan]

Seeliger's paradox is a long known one. Infinite mass, distributed over the infinite space.

The Hurkyl example is of a finite mass, distributed over the infinite space, but it's not a paradox (yet). Only the mass center has no finite distance form 0.

Bad enough, if you ask me, but not my problem.

I have WELL defined mass center in my example. At the finite distance - 10/19.

And still, everything goes south.

 

[vanesch]

And still, everything goes south.

No, you don't even know if everything goes south ! It depends on how you make your sum. But if you say that FIRST summing all the forces on one single particle is the way to go (which is a refinement of the superposition idea in Newton's laws), then yes, everything goes south and that is not even a contradiction. Why ? Because the proof of conservation of momentum in a closed system is simply not valid anymore. In the proof, you sum the forces 2 by 2 on different particles. Clearly, that's not to be done if you first have to sum them all on one particle. The two results are only the same if "taking the sum" is insensitive to the order of summation, which is not the case here. Hence, it is not justified to take the forces two by two, and hence the proof of the conservation of momentum is based upon a non-founded hypothesis (we are allowed to alter the order in which we sum the forces, which is clearly not the case here).
As such, there is no conservation of momentum in this example. So what ?
For the system you specified, with the refinement of how to sum the forces (which is needed to even know what is the "sum"), there is no conservation of momentum. That's not a contradiction, is it ?