A paradox inside Newtonian world

[Tomaz Kristan]

So how about this: Any supposed physical situation that reduces to a conditionally convergent series is unphysical.

Something should be done. Maybe it would be enough.

For now, the construction I gave, is perfectly legal.

 

[Tomaz Kristan]

An illustration of the problem. An illustration, nothing more.

The complex on the left side consists of the infinite number of ever smaller (1000 times by volume, 2 times by mass) spheres.

http://www.critticall.com/alog/Complex_of_Mass_Spheres.jpg

 

[ObsessiveMathsFreak]

There is nothing (almost nothing) to calculate. The surface reaction forces are defined to balance any negative gravitational net force.

Which part of;
L_{n} = L_{n+1} - G_{n+1} +m_{n+1} \frac{G_{n+2} + L_{n+1} -L_{n+2}}{m_{n+2}}
is giving the trouble exactly. To me, it''s fairly straightforward that this equation says that the left reaction force on any ball depends on the left reaction forces of the two balls to the left of it. Would you care to comment.

They are not defined to balance any positive (repulsive) gravitational net force in this case. They could also be, but they are not.

You do realize that what you're saying means that whatever model you are working with, is totally unrealted to the original problem. You've introduced some external holding force on each ball, justified by nothing more than your own handwavings.

The paradox arises, if the R^3 set and the Newton's axioms are combined. Who is "to blame", I don't know.

I'm putting the balme squarely with the "lack of rigor" department.

I know, it's not a real world problem, it's just a problem of an axiomatic system, we believed it was sane.

It's a mathematical problem. So far, you've provided close to zero mathematics, Now either you're going to provide some solid mathematics that everyone can digest, or you're going to keep obstinately refusing to even put down one equation. Dodgeing the issue with handwavings or dismissing valid arguments completely smacks of crackpottery.

I'm not often this caustic, but frankly I feel at this point that you are deliberately wasting everyone's time here.

Edit:
P.S.
Please turn that bmp into a jpeg. You're causing an internet traffic jam.

 

[Tomaz Kristan]

Why two? Why this equation at all? Who invented it? You?

I think, that EACH ball on the left side has its gravitational pull to a ball. All those pull forces are to be added together.

When we do that, we still have a finite force, canceled out by the surface reaction force.

Plus, a finite force from the right side is also always present, always finite.

What more do you need? The gravitational force formula? Some infinite series sums?

 

[ObsessiveMathsFreak]

Why two? Why this equation at all? Who invented it? You?

I derived it in post 136!

What more do you need? The gravitational force formula? Some infinite series sums?

The answers Kristan! Give your answer for the accelleration. It's the result of mathematical equations. Show your equations. The equations you presumably came up with when you first investigated these paradoxes so that you would at least have some idea what you were talking about when you presented them here rather than have the rest of us go to the trouble of coming up with just about everything for you.

Honestly!

 

[Tomaz Kristan]

The answers Kristan! Give your answer for the accelleration. It's the result of mathematical equations.

I can't believe this.

F(N,N+1)=G*m(N)*m(N+1)/(10^(N+1))^2=G*m(N)*(m(N)/2)/(10^(N+1))^2

For every N.

Do you agree so far?

 

[HallsofIvy]

You can't believe what? That people would ask you to verify what you say? That people would ask you to show exactly how you arrived at the equations you use? What you've given is the gravitational force between two consecutive masses. What is the total force on each mass and what is its acceleration?

 

[Tomaz Kristan]

You can't believe what?

That you don't find this arithmetic trivial.

What is the total force on each mass and what is its acceleration?

Okay, if you want it the slow way.

Do you agree or not:

F(N,N+1)=G*m(N)*m(N+1)/(10^(N+1))^2

or

G*m(N)*(m(N)/2)/(10^(N+1))^2

or

G*(m(N)^2/2)/(10^(N+1))^2


For every N.

 

[Tomaz Kristan]

In the case you agree, do you agree with:

F(N,N+1)+F(N,N+2)+F(N,N+3)+ ... < 2*F(N,N+1)

Since the m(N+1)=m(N+2)+m(N+3)+m(N+4)+...

and the mass is actually distributed more away, than if it was all squeezed to the immediate left ball.

 

[ObsessiveMathsFreak]

In the case you agree, do you agree with:

F(N,N+1)+F(N,N+2)+F(N,N+3)+ ... < 2*F(N,N+1)

That came out of nowhere, again without any proof whatsoever.

Since the m(N+1)=m(N+2)+m(N+3)+m(N+4)+...

and the mass is actually distributed more away, than if it was all squeezed to the immediate left ball.

And...? What relevance has any of this to the accelleration of the system? You still haven't given any answers yet.

 

[Quaoar]

I hate to repeat myself but what's wrong with saying that conditionally convergent series are unphysical? Doesn't this resolve the issue?

 

[vanesch]

I hate to repeat myself but what's wrong with saying that conditionally convergent series are unphysical? Doesn't this resolve the issue?

The problem is not with the mathematics. The problem is with the axiomatic system of Newtonian mechanics, which allows (if not explicitly forbidden) a setup in which, in order to calculate the force on a body, the rule that is given by the axiomatic system LEADS TO a conditionally convergent series.

Imagine a force law which contains a term : sqrt(d - 5) where d is the distance between two points. You can now say: the fact that you get difficulties for d < 5 is related to saying that imaginary numbers are unphysical. Right. But the problem is that if the rule said that you had to apply sqrt(d - 5), and for the specific setup at hand this gives you an imaginary number (because d = 3 for instance), then that's not a problem of mathematics but of the force law at hand: it must make sure that d cannot be smaller than 5. And in the example cited here in this thread, there is no a priori reason why it should not occur (except for diverging volume mass density).

 

[Quaoar]

The problem is not with the mathematics. The problem is with the axiomatic system of Newtonian mechanics, which allows (if not explicitly forbidden) a setup in which, in order to calculate the force on a body, the rule that is given by the axiomatic system LEADS TO a conditionally convergent series.

My point is that any problem that results in a conditionally convergent series couldn't be constructed in the first place. In this case, it's because we have an infinite number of masses.

 

[Tomaz Kristan]

OMF,

One of this days I'll tell you, what value exactly the sum of all left forces are. I assure you, it is smaller than if two immediate left balls were in the same place. I assure you!

My point is that any problem that results in a conditionally convergent series couldn't be constructed in the first place. In this case, it's because we have an infinite number of masses.

A good suggestion for the future improvements, maybe. But it is not on the table now. Nor has Newton forbid this crap, not is forbidden in the R^3 set.

vanesch,

Thank you for understanding the whole picture. I like your remark, that the system might produce this kind of situation by itself, calculating or simulating something else, completely harmless at the first glance.

 

[vanesch]

My point is that any problem that results in a conditionally convergent series couldn't be constructed in the first place. In this case, it's because we have an infinite number of masses.

Well, yes, but the total mass is finite, so this simply comes down to a special distribution of a finite mass over space. If this is not allowed, then we could not do Newtonian mechanics in continuum mechanics, where we chunk up a finite mass in an infinite number of infinitesimal amounts of mass. So it can not be this infinite chunking up which gives problems by itself.
I have my gut feeling that it is the divergent mass density which is the culprit, but there are 2 caveats:
- one should then have to be able to show that in cases where there is no such divergent mass density, that a conditionally convergent series cannot occur
- one should also (that's way trickier !) have to show that such situations cannot evolve, under Newtonian dynamics, from much more innocent mass distributions which do have finite mass densities.
(because in that way, these more innocent initial conditions should also have to be forbidden etc... and, by Poincare, we might end up by forbidding 99.9999% of all of phase space as initial condition!)

Now, as already said, this doesn't mean that there is any practical problem with Newtonian physics as an effective theory. But it is a surprise to me that it fails as an axiomatic system.

 

[vanesch]

You can't believe what? That people would ask you to verify what you say? That people would ask you to show exactly how you arrived at the equations you use? What you've given is the gravitational force between two consecutive masses. What is the total force on each mass and what is its acceleration?

In earlier posts I attached some mathematica notebooks calculating numerically the forces to the first problem which was posed in this thread. The results were indeed only numerical but I think it is not difficult to establish analytic inequalities.

 

[Tomaz Kristan]

I have my gut feeling that it is the divergent mass density which is the culprit

As a matter of fact, there is a certain possible way, to flatten those balls to pancakes. To avoid arbitrary big densities.

 

[Tomaz Kristan]

OTOH, those close pancakes still cause the arbitrary high densities.

But it is conceivable, to have a force between those balls, which is proportional to r^-4. That way those balls could have even a constant density, still causing the paradox.

Anyway, something must go to arbitrary high (yet finite!) values in order to have this problem. It will remain so, until some maximal or minimal values aren't there. In the real physics we have the speed of light, Planck size and time, and so on, already.

 

[Hurkyl]

Nor has Newton forbid this crap

What axiomization of classical mechanics are you using?

 

[Tomaz Kristan]

What axiomization of classical mechanics are you using?

Newton's three laws of motion. The first is already a derivative, a theorem of the second, so the II. and the III.

Plus of course everything what comes with the so called real numbers.

Plus the gravity law, but this one could be replaced by some other force, behaving like that. Or even "better", to be proportional to r^-5 or something.

So, the second and the third Newton's law inside R^3 or even R.

Enough to get there, where a paradox lives.

 

[Hurkyl]

Newton's three laws of motion.
...
Plus of course everything what comes with the ... real numbers.

Newton's laws are rather informal, and certainly not complete. e.g. you've listed nothing that tells you, e.g., that a particle is something with mass and position. (which is a problem, because I don't think you can even state Newton's laws until you've postulated that particles have mass and position)

Before asking this question, I did a brief search for axiomatic Newtonian mechanics, and found Axiomatic foundations of Classical Particle Mechanics. Some key points about this paper are:

The first axiom is that there are only finitely many particles.

In theorem 3 (which deals with center of mass), they remark that the assumption of finitely many particles is essential to their formalism.

If we used their formalism, then there is no paradox: your construction is illegal.

If we took their formalism and threw out the axiom that says there are finitely many particles, then you don't have a center of mass theorem, and once again there is no paradox.

 

[Tomaz Kristan]

Page 258 bottom and 259 top:

But an essential generalization of our axiom system would be obtained if we were to replace P1 by the axiom: "P is nonempty, and either finite or countably infinite." If the axiom P1 were to be liberalized in this way, however, then it would probably be desirable to add some additional axioms, as to insure that the total mass and kinetic energy of the system be finite.

 

[Hurkyl]

I repeat::

If we took their formalism and threw out the axiom that says there are finitely many particles, then you don't have a center of mass theorem, and once again there is no paradox.

 

[Tomaz Kristan]

No. The mass center can be very well defined for the infinite number of particles, when the mass remains finite. At least.

 

[Tomaz Kristan]

The paradox does not live only with the infinite set of bodies. You may consider those left balls as glued together, and the Jupiter as the second body on the right side.

The paradox blooms fine, with "only" the infinite divisibility of the matter.

 

[vanesch]

The first axiom is that there are only finitely many particles.

In theorem 3 (which deals with center of mass), they remark that the assumption of finitely many particles is essential to their formalism.

If we used their formalism, then there is no paradox: your construction is illegal.

Yes, but so is continuum mechanics then...

That's the "problem": we use Newtonian mechanics regularly with an infinite amount of "mass points".

Of course, Newtonian mechanics, limited to two mass points, and non-zero total angular momentum, is an entirely consistent axiomatic system. With one mass point also

 

[Hurkyl]

Yes, but so is continuum mechanics then...

Right; that's why that paper is an axiomatic foundation for particle mechanics, and not for continuum mechanics. I, actually, would really like to see a more general axiomatic foundation, it's just that this was all I could find.

That's the "problem": we use Newtonian mechanics regularly with an infinite amount of "mass points".

Actually, if you use the techniques of nonstandard analysis, then these axioms are adequate. (you only need hyperfinitely many particles to approximate your continuum)

e.g. In Tomaz's original scenario NSA tells us that a particle of infinitessimal mass gets flung rightwards at transfinite speed, and that exactly makes up for the missing momentum. (That's why I was making a big deal about the behavior about the origin, because that's my best guess as to the standard analog)

 

[Tomaz Kristan]

e.g. In Tomaz's original scenario NSA tells us that a particle of infinitessimal mass gets flung rightwards at transfinite speed, and that exactly makes up for the missing momentum. (That's why I was making a big deal about the behavior about the origin, because that's my best guess as to the standard analog)

Every ball (no matter how small) has a much bigger left drag, than the right drag. It quickly escapes from the gravity of its right neighbor.

No right moving whatsoever!

 

[Hurkyl]

Every ball (no matter how small) has a much bigger left drag, than the right drag. It quickly escapes from the gravity of its right neighbor.

No right moving whatsoever!

In the nonstandard model we'd use to analyze your scenario, there's a leftmost ball.

 

[Tomaz Kristan]

In the nonstandard model we'd use to analyze your scenario, there's a leftmost ball.

There is NO leftmost ball at all.

 

[ObsessiveMathsFreak]

There is NO leftmost ball at all.

But there is in the limiting case as the number of particles goes to infinity.

 

[Tomaz Kristan]

Limit for what? For the "leftmost" ball speed after a second? For the force between the two "leftmost" balls?

Nothing like that exists.

 

[ObsessiveMathsFreak]

Limit for what? For the "leftmost" ball speed after a second? For the force between the two "leftmost" balls?

Nothing like that exists.

In the limit as the number of balls goes towards infinity. As we logically increase the number of balls, in each of our cases, there is a leftmost ball.

 

[Tomaz Kristan]

In the limit as the number of balls goes towards infinity. As we logically increase the number of balls, in each of our cases, there is a leftmost ball.

No, it doesn't go that way and you know that.

 

[Hurkyl]

There is NO leftmost ball at all.

There is, in the nonstanard model. It contains H balls, where H is a transfinite (hyper)integer. The H-th ball is the leftmost.

 

[radou]

This discussion seems to be an ideal example of an infinite process.

 

[Gelsamel Epsilon]

What does transfinite mean? Boundless but not infinite?

 

[Hurkyl]

The standard natural numbers form an (external) subset of the hypernatural numbers. We (externally) define that a hypernatural number is transfinite if and only if it is larger than every natural number.

The word "transfinite" is used to distinguish it from the standard usage of "infinite", since, for example, a transfinite sum is something different than an infinite sum. (But their values are infinitessimally close, if the summand is well behaved)

 

[Gelsamel Epsilon]

So basically transfinite numbers are numbers which are larger then any finite number but smaller then infinity?

 

[Hurkyl]

So basically transfinite numbers are numbers which are larger then any finite number but smaller then infinity?

That is an accurate statement.

(resisting urge to go into what is probably unnecessary detail)

 

[Tomaz Kristan]

So, you say, that you are going to clean up the mess by adding some more infinite stuff?

Well, maybe, who knows, but currently those transfinite shadow balls are nowhere defined, inside the Newtonian world. It's still to be done, if it's of any use, anyway.

Now, it's no solution.

 

[Gelsamel Epsilon]

That is an accurate statement.

(resisting urge to go into what is probably unnecessary detail)

Go into more detail if you really want too, I'm always keen on learning things.

 

[Tomaz Kristan]

Don't steal me my thread, please. Go elsewhere. Unless he somehow solve the paradox I gave, with those hypernaturals. Hypernumbers deserve a new topic. Here are welcome only iff something become clear using them.

 

[ObsessiveMathsFreak]

No, it doesn't go that way and you know that.

It does go that way. That's what an infinite sum means.

When we write \sum_{n=0}^{\infty} a_n, what we mean is;

\lim_{k \to \infty } \sum_{n=0}^{k} a_n

It's clear as crystal. In each limiting case, there is a leftmost ball and the center of mass remains fixed. In the limit as k \to \infty, the accelleration of the center of mass is zero. And that is all we can say without progressing to very esoteric arguments about things like hyperreal numbers etc.

 

[Tomaz Kristan]

You argument is false, OMF.

If it hadn't been, you could just as well proved, that the biggest natural number exists. Bigger than any other.

Well, it's a basic mistake on your side, trust me!

 

[vanesch]

It does go that way. That's what an infinite sum means.

When we write \sum_{n=0}^{\infty} a_n, what we mean is;

\lim_{k \to \infty } \sum_{n=0}^{k} a_n

It's clear as crystal. In each limiting case, there is a leftmost ball and the center of mass remains fixed. In the limit as k \to \infty, the accelleration of the center of mass is zero.

Yes, this is correct. It is because you approach the final situation as a sequence of situations in which there are each time a finite number of balls, but more and more of them. However, there's no guarantee that this "adding balls" is the one and only correct "limiting procedure".

The other approach, the one that leads to the paradox, is, by NOT setting up a sequence of situations with more and more balls, but by considering all balls at once, and calculate the total force on each individual ball. If you do that, it turns out that the sign of the force on each individual ball, by the entire set of all balls, is the same.
Adding now the forces of all balls together will then result of course in a net force with the same sign.

There are similar situations where you cannot just consider sequences of physical setups, and take the limit of a quantity in this sequence, as the value of the quantity that would occur in the limiting situation. Another example is this:

Consider an Euclidean space with a homogeneous, constant mass density. Turns out (by symmetry) that this mass density doesn't result in any (Newtonian) gravitation force on a test mass.
However, if you approach this situation by considering a sphere of radius R with homogeneous mass density, and 0 outside, then your test mass will undergo, for each value of R, a specific force towards the center of the sphere. If R > d (distance between test particle and center of sphere), then this force will not change anymore. So the force, as a function of R, grows first, and becomes a constant from the moment R > d. Taking the limit R - > infinity gives you this constant force.

Nevertheless, the physical situation with R-> infinity is a space filled with a homogeneous mass density, where the force should be 0.

 

[ObsessiveMathsFreak]

You argument is false, OMF.

If it hadn't been, you could just as well proved, that the biggest natural number exists. Bigger than any other.

Well, it's a basic mistake on your side, trust me!

I assure you, an infinite number of non zero numbers sums to infinity. When we speak of "infinte" sums, we are in fact speaking about the limiting case as the number of terms in the sum increases without bound. That's what a "sum to infinity" really means. The limiting case.

 

[Tomaz Kristan]

> I assure you, an infinite number of non zero numbers sums to infinity.

1/2+1/4+1/8+ ... = 1

Don't you think so?

 

[Alkatran]

Important point that's probably already been made: the sum of an infinite sequence is equal to the limit of the sum of its first n terms as n goes to infinity. We need this definition to avoid problems such as:

This sequence converges
<br /> \sum_{n=1}^\infty \frac{(-1)^n}{n}<br />

However, because 1/n diverges as we sum to infinity, we can just rearrange the terms so that we get a series of the form

1/2 + 1/4 + 1/6 + 1/8 ... (add until we reach at least 3) - 1/1 + ... (add until we reach at least 6) - 1/3 + ...
As you can see, this series diverges to infinity, even though it as the exact same elements as our alternating converging series. The order we sum the elements in makes a HUGE difference!

Therefore, to calculate the value of your system, you HAVE to take the limit as n approaches infinity, which means you always consider the left most (nth) ball, your forces cancel, and the problem disappears.

 

[vanesch]

Therefore, to calculate the value of your system, you HAVE to take the limit as n approaches infinity, which means you always consider the left most (nth) ball, your forces cancel, and the problem disappears.

As I tried to point out, that's only one possible way to "approach the system": considering systems with n balls, calculating the total force on each ball in this n-ball system, sum the forces for the force on the CoG, and find 0 for each n. Then take n-> infinity for {0,0,0,...0,...} which gives 0.

But another possible way gives you the paradox. Instead of taking the sequence of systems with n balls with n -> infinity, we can also consider directly the system of infinite number of balls, but this time, calculate the total force on ball number k, F_k. F_k can be shown to be of same sign, no matter what k. We then consider the sum over the first n of these F_k, and take now the limit n -> infinity, to find the total force. And this time, the sum doesn't go to 0.

As you point out, this comes because we are re-arranging a conditionally-convergent series (but we already talked about that in the beginning of this thread). So yes, we do understand mathematically, how it comes that we get different results according to different approaches. However, Newton's axioms don't tell us which is the "right" approach.

One could think that "adding balls one by one" is the obviously correct way, because that's what makes the CoG forceless.

However, I gave a counter example (the space filled with homogeneous mass density) where it is this time this technique that fails (the growing sphere with R - > infinity doesn't give you the right force on a test particle), and where it is the "sum over forces" technique that gives you the right result.

So this means that, from case to case, in a Newtonian system, there are different approaches possible which give different answers, and it is not a priori clear which is the "right" answer. That's what is called an inconsistent axiomatic system.
(and again, as long as there is a finite number of balls, there's no problem, but we often use Newtonian mechanics outside of this scope).