A paradox inside Newtonian world

[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 ?

 

[vanesch]

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

I also had finite mass...

 

[Hurkyl]

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.

Wrong. It doesn't have a mass center.

If the sum isn't pathological enough for your tastes, then you can put a mirror image of my configuration on the negative side of the x-axis.

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.

Why?

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

And why, pray dell, do you think you are above such considerations? Why don't you show us the law which permits constructions like yours?

But you're missing half the point -- it's not just that you need to show your configuration is legal, but you also need to show that the center of mass theorem applies to your configuration.

Some versions of classical mechanics, such as the one I've already posted in this thread, explicitly forbid your mass configuration.

Other versions of classical mechanics might permit your configuration, but not insist that the center of mass theorem applies to it.

Other versions (the one you seem to have in mind, and steadfastly refuse to consider that things might be any other way) would permit your configuration, insist the center of mass theorem applies to it, and wind up being logically inconsistent. And thus most people wouldn't bother giving it a second thought. (And most of those who would only do so to study why it doesn't work)


laws[/url] do not make any statements about what mass configurations are legal, and they don't say a single word about the center of mass.

 

[Hurkyl]

Note Godel says that paradoxes cannot be avoided.

He did? His famous theorems don't assert such a thing.

 

[Tomaz Kristan]

Other versions of classical mechanics might permit your configuration, but not insist that the center of mass theorem applies to it.

I was under the impression, that you don't see an obvious thing. And as I see now, vanesh also doesn't.

Take a pyramid instead of my initial construct! Then you can imagine, how its points migrate (under system's internal forces only) the way, the initial construct grew.

Morphing of a kind.

Had the construct was immune to the momentum conservation theorem, the pyramid would also be!

What I have nothing heard about.

See now?

 

[reilly]

Hurkyl -- As you must know by now, I'm a very simple minded guy. All I mean to say is that there are questions that cannot be answered. Yes, I'm fully aware of all the intricacies of Godel's work -- I even understood it once, when I was much younger, in my 40s. So, having been in marketing a long time, I try to frame my ideas, as much as possible, in simple language, and, if possible, use a broad brush in order to mitigate highly technical issues.

Do I need Godel to support my notion? Probably not, but then people will pay more attention to the name Godel than to the name, Reilly Atkinson, who says, paradoxes abound, and are part of the human condition, a result of our neural wiring -- which allows such paradoxical states as humans laughing and crying at the same time, humans can love and hate at the same time -- talk about superposition. We are finite beings who can ponder the meaning of infinity, which does not appear in nature, but only in our imperfect heads. We really don't need Godel to point out huge problems with our languages and rationality. By the way, how high is up?

....
TK -- So what's the story on conditional convergence? I hope that your
silence is not telling.

Regards,
Reilly Atkinson

 

[StatusX]

To expand on what I said in post 273, here's one way to formalize Newton's theory:

We define a system as a set of point masses at positions x_\alpha(t) \in R^3, where t \in R and with masses m_\alpha \in (0,\infty), indexed by a set A (ie, \alpha takes on values in A, which could be, eg, the natural numbers). Then we assign to each pair (\alpha,\beta) \in A^2 an interaction force f_{\alpha \beta}(t) \in R^3. Then the two main axioms of Newtonian mechanics are:

1. m_\alpha \frac{d^2 x_\alpha}{d t^2} = \sum_{\beta \in A} f_{\alpha \beta}(t)

2. f_{\alpha \beta}(t) = - f_{\beta \alpha}(t), \mbox{ all } t \in R

So far we have made no other restrictions on the parameters of the model, which to summarize, are:

m_\alpha, x_\alpha(t), f_{\alpha \beta} (t), A.

For (1) to be well defined, the sum on the LHS should converge absolutely (since we haven't specified an order to take the sum). So we require:

3. \sum_{\beta \in A} ||f_{\alpha \beta}(t)|| < \infty, \mbox{ all } \alpha \in A, t \in R

At this point we potentially have an infinite number of coupled differential equations, and I'm not familiar with the theory of such equations. The model might already be inconsistent. But we can forget about actually solving these equations for now and just treat d/dt as an arbitrary linear operator (ie, we'll drop t dependence and just look at this as a fixed configuration of masses and forces).

Next let's consider a situation where the only force is gravity, ie, we define:

f_{\alpha \beta} = \frac{G m_\alpha m_\beta}{||x_\alpha - x_\beta||^3} (x_\beta- x_\alpha)

Which satisfies (2) (assuming all point masses are at distinct positions). For the configuration in post 1, (3) is also satisfied, as is easily checked. So we see that that system is perfectly valid under this model, which we'll call (1,2,3).

The question is whether conservation of momentum holds in this model. Specifically, if we define (of course, now we need to limit to systems where these are defined, but they are for the system in question):

M_{tot} = \sum_{\alpha \in A} m_\alpha

x_{CM} = \frac{1}{M_{tot}}\sum_{\alpha \in A} m_\alpha x_\alpha

p_{tot} = M_{tot} \frac{d x_{cm}}{dt} = \sum_{\alpha} m_\alpha \frac{d x_\alpha}{dt}

(where we have freely interchanged the derivative with the infinite sum, which again may not be valid, but turns out not to be the problem) then we want:

\frac{d p_{tot}}{dt} =0

The way this is usually proved is to write:

\frac{d p_{tot}}{dt} = \frac{d}{dt} \sum_{\alpha} m_\alpha \frac{d x_\alpha}{dt}= \sum_{\alpha} m_\alpha \frac{d^2 x_\alpha}{dt^2}=\sum_{\alpha} \sum_{\beta} f_{\alpha \beta}

If at this point we impose the following axiom (strictly stronger than (3)):

4. \sum_{\alpha} \sum_{\beta} ||f_{\alpha \beta}|| < \infty

Then the sum above converges absolutely, so we can rearrange it so that f_{\alpha \beta} sits next to f_{\beta\alpha}, these will cancel by (2), and we have proven conservation of momentum in the model (1,2,4).

However, the system in post 1 does not satisfy (4), so does not fit into a model where conservation of momentum holds.

 

[vanesch]

Take a pyramid instead of my initial construct! Then you can imagine, how its points migrate (under system's internal forces only) the way, the initial construct grew.

Morphing of a kind.

Yes, and what is the problem with that ?

Had the construct was immune to the momentum conservation theorem, the pyramid would also be!

What I have nothing heard about.

See now?

Yes, but what is the problem ? We have a system about which the usual application of Newton's laws generate a set of forces which makes up a conditionally convergent series. This means that the value of its sum is undefined, and you have to add an extra rule to say HOW we should sum them in order to have a defined value for the forces to be filled into Newton's second law. Nice. You say that we should sum them in the order of the numbering of the points, on one point at a time. If we do that, then we don't have the conservation of momentum theorem anymore (which sums the forces in another order). And in a universe where that holds, yes, piramids of your construction morph as you describe. So what ?

 

[vanesch]

If at this point we impose the following axiom (strictly stronger than (3)):

4. \sum_{\alpha} \sum_{\beta} ||f_{\alpha \beta}|| < \infty

Then the sum above converges absolutely, so we can rearrange it so that f_{\alpha \beta} sits next to f_{\beta\alpha}, these will cancel by (2), and we have proven conservation of momentum in the model (1,2,4).

However, the system in post 1 does not satisfy (4), so does not fit into a model where conservation of momentum holds.

Yes. Very nice formalisation. Now, the tricky thing is: if a system is satisfying axioms 1,2,3 and 4 at time t_0, will it satisfy them at time t_1 under the usual evolution ?

 

[reilly]

StatusX -- vanesch put it well, nicely done indeed. Am I mistaken that once you've got 1-4, the standard theory of(systems of) differential equations tells you that given the state at some time, then the system is specified for all time?

Regards,
Reilly

 

[Tomaz Kristan]

However, the system in post 1 does not satisfy (4), so does not fit into a model where conservation of momentum holds.

A pyramid with the uniform density would satisfy (4) clearly, and the momentum would be conserved.

Internal forces in the pyramid could rearrange it to become the original construct.

Therefore, the conservation of momentum does not hold for pyramid!?

and THAT is (another form of this) paradox.

 

[vanesch]

A pyramid with the uniform density

The piramid you constructed was not of uniform density, no ? Its density diverged at its vertex ?

 

[Tomaz Kristan]

The piramid you constructed was not of uniform density, no ? Its density diverged at its vertex ?

Of course, it was of uniform density. But it's a tree apple case, where the tree wood floats, apples grown later don't.

A pyramid--my construct transformation, needs only internal forces to come about.

To the pyramid become "momentum nonconservative", needs only internal forces. Therefore already is "momentum nonconservative".

We can't afford to have some finite shapes or forms, for which momentum conservation doesn't apply. Since a transformation to such, is just too easy.

 

[StatusX]

Vanesch and reilly - thanks. Both of your questions concern the time evolution of the system. I realized halfway through writing that post that considering that would make the model much more complicated to develop. For example, consider the system of coupled differential equations:

\frac{dx_1}{dt}=k_1 x_2

\frac{dx_2}{dt}=k_1 x_3

\frac{dx_3}{dt}=k_1 x_4

...

I don't even know if that's well defined, but even if it is, I couldn't tell you much about the the values at some later time or the uniqueness of a solution. I decided to not worry about this aspect, just treat d/dt as an arbitrary linear operator, and focus on proving d/dt(p)=0, ie, treating this as a geometrical rather than dynamical problem. It is an interesting question though, and I'll give it some more thought.

Tomaz, what is this pyramid you're talking about?

 

[Tomaz Kristan]

Tomaz, what is this pyramid you're talking about?

It's just an example of a body, which should behave well. Obeying Newton's laws. On one hand.

On another hand, a quick transformation, using only internal forces, reshapes it into my initial complex, or into the ball case, described later.

IFF my complex in nonNewtonian, a pyramid (cube, ball ...) - is also!

 

[StatusX]

I mean specifically what is the configuration? If you already described it please point me to the post.

 

[Tomaz Kristan]

Mass point (or ball) at 2^-N kg at 10^-N meter.

For every natural N.

Post number 1 has a link to a pdf file, where the complex or construction is described.

 

[StatusX]

I was talking about the pyramid you mention here:

A pyramid with the uniform density would satisfy (4) clearly, and the momentum would be conserved.

Internal forces in the pyramid could rearrange it to become the original construct.

Therefore, the conservation of momentum does not hold for pyramid!?

and THAT is (another form of this) paradox.

If you think you've found another paradox, or don't think I've settled the original one, please explain yourself more clearly.

 

[Tomaz Kristan]

You haven't settled the original one.

Even, if there are some nonNewtonian shapes, for which the conservation of momentum doesn't hold, as you claim, one can transform a pyramid (or something else) to this nonNewtonian shape of mine, using only internal forces - and therefore a humble pyramid is a nonNewtonian shape also.

Do you agree with me?

 

[StatusX]

If that's true, then you're right, it's not settled, and I'd have to work out how time evolution works, like mentioned above. I'll work on that. Do you have a specific process in mind that would transform a pyramid into the shape from post 1?

I mean, so far you can define f(t) to be anything you want, so for example, taking the original system from post 1, with f(t)=0 for t<0 and f(t) as from gravity for t>0, then you have a system satisfying (4) before t=0 and not after (or at least, not at t=0). So some more conditions are definitely going to need to be enforced to ensure things evolve nicely in time.

 

[disregardthat]

Sorry, this does not have anything special to do with the topic, i just don't know where to put it.
It is constated that a heavy object does not fall faster than a lighter object. If you think about it, it does. The heavy drags the Earth to itself in a way, right? the small object too, but since it is smaller, it does not drag it with the same "energy" as the heavy. So, wouldn't that say that heavy objects fall faster, although the difference is unmeasurable if you use the earth?