A paradox inside Newtonian world

[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?

 

[Tomaz Kristan]

Do you have a specific process in mind that would transform a pyramid into the shape from post 1?

Yes. For example, the 2.0kg pyramid is first sliced to 1.0kg, 0.5kg, 0.25kg and so forth pieces. Then each of those pieces is condensed to a mass point in appropriate places on the pyramid's axe.

Quite a complicated way to describe it mathematically, but all that can be regard as the work of the internal forces only.

Any internal mass transport is a finite one. For a finite distance in a finite time, in this case.

What makes the pyramid as nonNewtonian, as the target construction is.

 

[Tomaz Kristan]

And there is the problem. If the momentum is not conserved for my construct, than it is possible, that sometimes isn't for quite ordinary bodies.

Of course, in the abstract Newtonian world, only.

 

[vanesch]

Then each of those pieces is condensed to a mass point in appropriate places on the pyramid's axe.

One should then examine whether the forces which contract the piramid slices, do not suffer themselves from the same kind of divergence as the one we're trying to ban.

In other words, the thing that comes back, is: can we find a set of conditions which avoid these kinds of problems (appearance of a set of conditionally convergent forces on the whole system), and which are invariant under time evolution and which aren't so drakonian, that we eliminate in doing so, a good part of the practical applications of Newtonian theory too ?

As has been pointed out before: limiting the mass distribution to a finite amount of mass points, is ok, but this kills continuum mechanics.

 

[arildno]

I really don't see see what is so very problematic here.
After all, a continuum mechanics is justified as long as the problems we're looking at has certain characteristic "scales". Beneath some scales, for example on problems where extremely small lengths are relevant, the continuum hypothesis has ALWAYS failed, and will always fail, since the world isn't a continuum after all.

In the start example of OP, however, he has presented a problem that has no characteristic length scales; significant interactions and forces appear on ALL possible scales. Thus, I don't see that these paradoxes in any way put any NEW limits of validity/applicability on, say, continuum mechanics.

 

[Tomaz Kristan]

It's a formal problem. An error was hidden for 300 years. A paradox nobody was aware of.

We will fix it, no doubt about that. This way or another. But for now ... we are coming together to recognize it. I am glad.

 

[Hurkyl]

In what are you interested, here? Are you interested in a mathematical discussion where we consider how classical mechanics might be rigorously formalized? A historical discussion where you try to prove you are the first person to notice anything like this? Or a philosophical discussion where you try and show that physicists have been practicing classical mechanics in error?

These are three radically different discussions, and it would be good to know which you actually want to discuss. (And it would be nice to keep on that particular topic)

 

[Tomaz Kristan]

I wonder, if there is a hole in my quest to find a hole in the Newton's abstract system.

Looks like not. And that there _is_ an antinomy after all. The conservation of momentum has a weak point.

I know, that most people do not agree. I know that. Some are even furious, maybe a bit hostile.

But still, my construction is here and I am open to see, if I made a mistake. I am afraid, I haven't.

 

[Hurkyl]

Er, does that mean you want to have

a philosophical discussion where you try and show that physicists have been practicing classical mechanics in error​

and, in fact, are not interested in having

a mathematical discussion where we consider how classical mechanics might be rigorously formalized?​

 

[Tomaz Kristan]

NO. I could have both of them. For the second, I have a suggestion. Get rid of infinity. Not only infinity, but also too large finite numbers.

 

[Hurkyl]

Okay.

Let's start with something smaller than your assertion -- can you give an example of anyone actually making the mistake of applying the center of mass theorem outside its domain of validity?

 

[Tomaz Kristan]

In practice, there may be no problem. In practice. Since we don't use numbers above some finite value.

But who knows. Anyway, it's not good as it is. The Russell's set of all sets, which do not contain themselves - had no influence to the geometry in practice, but the Set theory had to be fixed non the less.

Once you have a paradox in the system, you can prove all you want - and all you don't want to prove, also.

Intolerable situation.

 

[vanesch]

Looks like not. And that there _is_ an antinomy after all. The conservation of momentum has a weak point.

The conservation of momentum by itself is not a problem: clearly for certain systems it doesn't hold. That's not a problem in itself. In fact, you haven't really found a contradiction in Newtonian physics, you've found a situation which is left undetermined, and which is quickly turned into a contradiction if we add arbitrary rules of how to lift the undeterminacy.

In your original proposal, the equations of motion for each individual particle are - I take it - defined, and the abstract total sum of all forces on all particles (which is the force on the COG) gives a conditionally convergent series. In other words, the total force is undetermined, and conservation of momentum doesn't hold in that case.

What is characteristic for such a situation is a divergence of the volume density.

In your piramid example, you show how a finite volume density can potentially evolve into the same situation, using a contrived set of forces, but which behave analytically correctly.

So what you've shown is that innocent-looking initial conditions and force laws (with enough bad faith) can hit a situation from which the further evolution is undetermined, or from which conservation of momentum doesn't hold anymore.

That, by itself, is not a logical contradiction. In the examples you gave, I think that the equation of motion for each individual particle is actually well-defined, it is only conservation of momentum which breaks down. There is no logical requirement for conservation of momentum, however. Conservation of momentum only holds when action = reaction, and when the total system of forces is absolutely convergent. Given that the system of forces in your examples doesn't satisfy this, well, for those systems, there's no conservation of momentum. So what ?

But things could even be worse: the equations of motion themselves could become ill-defined, if the force system on an individual particle were to be conditionally convergent. In that case, still, there is no contradiction. The only conclusion would be that the laws of Newton do not define an equation of motion beyond that point. Again, you could obtain a logical contradiction if you introduced different ways of summing that series.

 

[Hurkyl]

In practice, there may be no problem.

Then how can you assert that people have been practicing classical mechanics wrongly?

There are several nuances to your assertion with which I take issue.



The first, and foremost, is that you've not demonstrated that there is a logical contradiction.

(*) the center of gravity of anybody cannot be accelerated, unless some external force is applied to it​

Your argument assumes, among other things, that (*) is universally valid, and derives a contradiction. You've not attempted to prove that your assumptions are actually valid in any formulation of classical mechanics.


To wit, I've looked at two references (that PDF I've linked, and my undergrad physics textbook), and your assumptions are invalid in both. They both derive (*) under the hypothesis that the sum of the internal forces converges absolutely (to zero). In the PDF, (*) actually is universally valid, but only because the axioms forbid situations where the sum of the internal forces might not converge absolutely. (In particular, they permit only finitely many particles)



The next issue is how physics is practiced. Classical mechanics generally isn't studied as the formal consequences of an explicit list of axioms (Though I would like to see such a formulation!). Physicists aren't irrevokably tied to a particular formal system -- in fact, they tend not to bother writing one down. When something goes wrong, that doesn't mean physics has collapsed; it just means that you applied something when you shouldn't have. (And if anyone gets around to formalizing physics, then what you did will clearly have to be forbidden)

Even if classical mechanics was a formal system, and you did find a contradiction within it, then that simply means they need to tweak the formal system to eliminate the contradiction and preserve as much existing physics as possible. (and this part is analogous to the set theory example) If most arguments still work in the fixed system, then you can't really say they were in error, can you?

 

[StatusX]

I think the best way to fix this is to limit to finite numbers of point masses. This would automatically satisfy the necessary constraints, and we even know it would evolve nicely since the governing set of equations would be equivalent to a single first order ODE in R^n for some n.

If this is too limiting, another option would be to impose that masses cannot get arbitrarily small. This doesn't seem to be a practical problem, and would force any infinite array of particles (say, an idealized model of a crystal lattice) to have infinite mass, and so things like total momentum and center of mass wouldn't be defined anyway. I'm not 100% sure about how this would evolve in time though.

Also note that these constraints would have no effect on continuum mechanics. That is a completely different theory, and would be developed from separate axioms entirely, not as some kind of limit of point masses (even if it can be intuitively thought of that way). For the same reason, it doesn't make sense (at least in this model) to have a continuous distribution evolve into a set of point masses. Also note this would involve dissipating an infinite amount of energy.

 

[arildno]

Well, what so awful about fixing these pseudo-problems by just saying systems exhibiting merely conditional convergence in some form is unphysical and disallowed?