A paradox inside Newtonian world

[Tomaz Kristan]

Put the physics aside for a moment! Take just the Euclidean one dimensional space and the Newton's laws (including the gravity law) - and watch my example from the link in the post #1, as a pure mathematical construct.

Then ask yourself, how could it be, that we have only the negative forces, where's the 3rd Newton's law?

The thing is, that the negation of the 3rd law is already a theorem. My "pdf construct" leads to this conclusion.

 

[Eli Botkin]

Tomaz, I modeled this mass distribution and the gravitational forces between mass points, using Matlab software. The net force (sum of + and - forces) for up to 150 mass points was zero, as expected from Newton's laws. So although gravitational forces would result in all masses eventually coming together at the center of gravitation, there is no net change in momentum.

 

[Hurkyl]

Tomaz, I modeled this mass distribution and the gravitational forces between mass points, using Matlab software. The net force (sum of + and - forces) for up to 150 mass points was zero, as expected from Newton's laws. So although gravitational forces would result in all masses eventually coming together at the center of gravitation, there is no net change in momentum.

Have you analyzed the error you introduced by using only finitely many mass points?

I'll give you the punchline: at the very least, your model has a huge error in the force it predicts for its leftmost particle.

 

[RandallB]

Have you analyzed the error you introduced by using only finitely many mass points?

I'll give you the punchline: at the very least, your model has a huge error in the force it predicts for its leftmost particle.

How can you possibly have enough information about his Matlab model to draw such a conclusion about such a specific part of it, and conclude he’s done something wrong? Are not the results as expected?

 

[Tomaz Kristan]

Read the pdf link at the post #1, RandallB and do some deep thinking and you'll see, what situation we have here.

 

[Eli Botkin]

Hurkyl, a zero net force is obtained regardless of the number of mass points included in the model. This would be expected as a result of Newton's 3rd law. For ANY two mass points, A and B, the gravitational force that A exerts on B is the negative of the force that B exerts on A. Therefore, summing over all forces for all mass points, the result would still be a zero net force.

A zero net force implies a zero net momentum change, so the motion of the center-of-mass is unchanged in whatever frame you select for observation.

 

[RandallB]

Read the pdf link at the post #1, RandallB and do some deep thinking and you'll see, what situation we have here.

I did that over a month ago when this problem was easily solved by more than one science advisor, all Eli did was work out another formal demonstration. What I can't figure out is why so many smart people are still kicking this dead horse, and making this silly thing such a long thread.

 

[Tomaz Kristan]

The horse is not dead and it kicks back.

The paradox hasn't been solved. Not at all.

 

[Hurkyl]

For ANY two mass points, A and B, the gravitational force that A exerts on B is the negative of the force that B exerts on A. Therefore, summing over all forces for all mass points, the result would still be a zero net force.

That does not follow mathematically: additional assumptions are required. One example of a sufficient condition is for the sum of the forces to be absolutely convergent, but Tomaz's scenario doesn't satisfy that condition.

The paradox hasn't been solved. Not at all.

Of course it has -- the self-contradictory (and unwarranted!) assumptions you have made have been explicitly pointed out to you many times in this thread.

How can you possibly have enough information about his Matlab model to draw such a conclusion about such a specific part of it, and conclude he’s done something wrong? Are not the results as expected?

I didn't say he did anything wrong -- his results are exactly what is expected from the model he created. I'm saying he did not analyze the quantitative (and qualitative!) differences between the actual problem and his model of the problem.

 

[Eli Botkin]

As you can tell from the number of replies I've made to date, I am new to this forum. I expected more scientific sophistication than currently exists on "sci.physics.reativity."

The argument comes down to: do you accept that the sum of +a and -a must be zero? Every pair of point masses in Tomaz's "paradox" gives rise to two forces that sum to zero.

Tomaz, if that doesn't end it for you, then we await your solution

 

[Hurkyl]

Eli: infinite summation is not infinitely associative. Probably the simplest demonstration is the following two infinite sums:

(1 + -1) + (1 + -1) + (1 + -1) + ... = 0,
1 + (-1 + 1) + (-1 + 1) + (-1 + ... = 1.

(Of course, 1 + -1 + 1 + -1 + ... doesn't even exist!)

Furthermore, any conditionally convergent sequence can be rearranged so that it sums to any value you want. See the Riemann series theorem.


The proof I have seen in classical mechanics that the center of mass of a system of particles remains unaccelerated when the net external force is zero requires you to rearrange the sum, and it rests crucially on the hypothesis that the two arrangements

sum over all particles (sum of all forces on that particle)

and

sum over all pairs of particles (sum of the forces they exert on each other)

have the same sum. In general, one has no reason to expect those sums to be equal unless they were absolutely convergent.


In other words, this theorem says:

"If the net external force is zero, and the sum of all internal and external forces is absolutely convergent, then the center of mass is unaccelerated"

Tomaz's mistake is assuming that this theorem says:

"If the net external force is zero then the center of mass is unaccelerated"



If you want to see the error in your 150 particle model, look at the net force on the 150-th particle.

Then, make a model with 151 particles, and look at the net force on the 150-th particle.

 

[Tomaz Kristan]

Tomaz's mistake is assuming that this theorem says:
"If the net external force is zero then the center of mass is unaccelerated"

And that _IS_ true. You know this or not, doesn't matter much.

 

[Eli Botkin]

Hurkyl:
After adding the 151st particle the net force on the 150th particle is much different than it was before the addition. Is that surprising to you?

In both cases, however, the sum of all net forces is zero, as it is for any truncation of the infinite set of particles. The inference, then, that the sum remains zero in the limit seems to me to be a physically reasonable one.

Recall that each individual interaction between 2 particles results in 2 forces that sum to zero. So regardless of the length of the truncated series, zeros are being summed: 0+0+0+0+...

 

[Hurkyl]

After adding the 151st particle the net force on the 150th particle is much different than it was before the addition. Is that surprising to you?

No; in fact, that's exactly the phenomenon I wanted you to see. That is the point of my comment on error analysis: your model gets the force on the last particle wrong by over 10^197 Newtons!

For comparison, the net force on the first 100 particles is merely 1.90 * 10^128 Newtons in magnitude. (Which, incidentally, is something your model can accurately compute)

Because of the significant error, you cannot assume your model correctly computes the net force on the first 150 particles.


Incidcentally, the correct value for the net force on the first 150 particles is roughly 1.50 * 10^198 Newtons in magnitude.

In both cases, however, the sum of all net forces is zero, as it is for any truncation of the infinite set of particles. The inference, then, that the sum remains zero in the limit seems to me to be a physically reasonable one.

It may seem reasonable, but that doesn't make it right. That's why this is a pseudoparadox: if we're not careful, our intuition can lead us to an entirely incorrect conclusion.

Recall that each individual interaction between 2 particles results in 2 forces that sum to zero. So regardless of the length of the truncated series, zeros are being summed: 0+0+0+0+...

Yes, I agree that the sum will be zero for any truncation -- any finite sum is absolutely convergent, and so we can rearrange and regroup its terms without changing the value of the sum.

But the relevant sum in Tomaz's problem is not absolutely convergent -- rearranging the terms can (and does!) change the value of the sum.

 

[Tomaz Kristan]

But the relevant sum in Tomaz's problem is not absolutely convergent -- rearranging the terms can (and does!) change the value of the sum.

Think again!

 

[masudr]

Think again!

Why?

*filler characters to increase the length of the post so the forum software will accept the post*

 

[Eli Botkin]

Hurkyl:
How would one rearrange the terms of the infinite string 0 +0 +0 +0 +0 +0 +... to change the value of the sum?
___________
Eli

 

[misnoma]

STOPPPPPPPPPPPPPPPPP: there is no paradox. Simply do the vector sum for all the masses involved with regards to their position. If the resultant of all forces is zero, then there will be no change in positin for any of the masses. If there is a change in position to satisfy equilibrium, then one, two or all of the masses will assume a position to satisfy the condition. Applying the rule that F = G (M m /r2) then all should be well.

 

[Hurkyl]

Hurkyl:
How would one rearrange the terms of the infinite string 0 +0 +0 +0 +0 +0 +... to change the value of the sum?
___________
Eli

We're not talking about an infinite sum of zeroes here. We're talking about the sum of all of the internal forces. You only get a string of zeroes when you rearrange and regroup this summation so that they pairwise cancel.



But, in the course of writing this post, I realized that it's a moot point -- in Tomaz's problem, the "force" on the center of mass cannot even be computed as a sum of forces on the individual particles.

Let's go back to first principles. First, some definitions:
x_i = the x-coordinate of the i-th particle. (1 \leq i)
m_i = the mass of the i-th particle.
x_c = the center of mass
m = the total mass
F_i = the net force on the i-th particle
F_c = the net "force" on the center of mass

By definition, the center of mass is given by

<br /> x_c = \sum_{1 \leq i} \frac{m_i x_i}{m}.<br />

(Furthermore, the definition of center of mass requires that this sum converges absolutely)


The "force" on the center of mass is

<br /> F_c = m \frac{d^2 x_c}{dt^2}<br />

If we plug in the definition of the center of mass, we get

<br /> F_c = m \frac{d^2}{dt^2} \sum_{1 \leq i} \frac{m_i x_i}{m}<br /> = \frac{d^2}{dt^2} \sum_{1 \leq i} m_i x_i.<br />

Under good circumstances, we can pull the derivative into the summation, giving us F_c = \sum_{1 \leq i} F_i. But this is not a good circumstance -- the theorem that would normally allow us to swap the order of summation and differentiation requires that

<br /> \sum_{1 \leq i} m_i \frac{d x_i}{dt}<br />

and

<br /> \sum_{1 \leq i} m_i \frac{d^2 x_i}{dt^2}<br /> = \sum_{1 \leq i} F_i<br />

both to be uniformly convergent (in t). But we don't have that here -- in fact, the latter sum doesn't even converge when t = 0.


This is all fine, since I'm pretty sure the center of mass doesn't exist for this collection of particles when t is nonzero, so it would be surprising if these sums were well-behaved.

 

[Eli Botkin]

Hurkyl:
I find no argument against the mathematical principles that you state. But we're trying to clear up what Tomaz thinks is a "paradox" in Newtonian physics.

First I would say that the choice of an infinite set of masses already takes us out of the bounds of physics per se since clearly there is no way to test predictions about such a set. So the only thing that remains, if we're to pursue the question at all, is to seek the limit, as masses are continually added, without limit, to what is initially a finite set of masses.

And (here I repeat myself) since in Newtonian physics a gravitational interaction between two massive particles always results in two forces that sum to zero, we are not unreasonable to "predict" that this would remain so even as the number of particles becomes unbounded. Accepting this allows me to "predict" that the sum of all forces generated by these multiple interactions would also be zero, and the motion of the particles would be such as to keep the sum of their momenta unchanged.
__________
Eli

 

[Hurkyl]

But we're trying to clear up what Tomaz thinks is a "paradox" in Newtonian physics.

Er, I'm slightly confused. Did you mean this post to be an attempt at reproducing the flawed reasoning that led Tomaz to his pseudoparadox? Or was this post supposed to be an attempt at explaining why it is not a paradox?

 

[Eli Botkin]

Hurkyl:
I don't know what it is that I wrote that confused you regarding my position visa vis Tomaz's "paradox." Tomaz concluded that the initially at rest CM would drift toward the smaller masses because the directions of the net force on each mass particle is toward the smaller masses.

But for any truncated set of masses this is not so, even though all forces except the one on the smallest mass are also directed toward the smallest mass.

Provided we stay in the realm of physics (a finite set of mass points) the masses will move in a manner which keeps their total momentum conserved. If the CM was initially at rest, it will remain at rest.

Tomaz's extrapolation to an infinite set of masses is outside of physics leaving us only to surmise what might happen. I feel free to surmise that in the limit you get the same result that applied for the truncated series of masses: no CM motion of an originally at rest CM. There is no paradox.
_________
Eli

 

[Hurkyl]

Tomaz's extrapolation to an infinite set of masses is outside of physics leaving us only to surmise what might happen. I feel free to surmise that in the limit you get the same result that applied for the truncated series of masses: no CM motion of an originally at rest CM. There is no paradox.

You can feel free to surmise, but that doesn't make you right. In fact, this supposition is one of the ways to arrive at the very pseudoparadox you're trying to avoid, because it contradicts the fact when you actually compute the center of mass, you find it does not exist as the system evolves.

You cannot resolve a paradox by ignoring half of it.

 

[Eli Botkin]

Hurkyl:
What do you see as the paradox? Newtonian physics predicts that the CM motion would be such that a system, not subjected to an external force, would conserve its momentum. Tomaz's infinite particle system isn't even within physics, let alone Newtonian physics, so how can it be claimed to be a Newtonian physics paradox? If you wish to say that Tomaz's system has no solution, I'm not unhappy with that.

I think we've beaten this to death. See you on some other (hopefully physics) question.
___________
Eli

 

[Hurkyl]

Let (*) denote the statement:

If there are no external forces on an arrangement of particles that has a center of mass, then the center of mass exists at all times, and moves with constant velocity.


The contradiction I see arises from assuming both:

(1) Tomaz's problem is a "valid" arrangement of mass points.
(2) (*) applies to Tomaz's problem.

From these assumptions, Tomaz claims to derive a paradox in the mathematical formulation of classical mechanics. I believe neither assumption is warranted.

As you point out, (1) is an unwarranted assumption because classical particle mechanics (typically) deals with only finitely many masses. (But, as vanesch points out, it is still interesting and useful to consider particle mechanics theories that permit infinitely many particles)

And, for the reasons I have been trying elaborating, (2) is an unwarranted assumption. There is no compelling reason to take (*) as an axiom of a generic infinite-particle theory of classical mechanics. Methods one might try to use to prove (*) turn out to fail -- the finite-particle proof of (*) depends on absolute convergence, and limiting arguments such as yours have similar problems.

 

[Tomaz Kristan]

To think, that we can't have more than a finite number of bodies in the Newton's world - is wrong.

Every cube, is a half of a cube, plus a quater of it ... and so on. Their interactions must be 100% covered by the Newton's laws.

 

[Hurkyl]

There aren't any cubes in particle mechanics.

 

[Eli Botkin]

Tomaz:
Imagining any extensive body to be made up of an infinite number of infinitesimal "bodies" may seem useful in some probems. But not in physics, where integration of the infinitesimal volume dxdydz seems almost invariably to have finite limits. True, we may conjure up an "infinitely long" wire in computing the magnetic field due to an electrical current. But this is known to be an idealization whose solution converges to a testable result. The system you posed is an idealization, but not of a testable physical system.

Tomaz, it's an interesting and instructive problem (as indicated by the length of the thread that you generated), but I remain unconvinced that it is a "paradox" in Newtonian physics.
____________
Eli

 

[Tomaz Kristan]

There aren't any cubes in particle mechanics.

Who says "particle mechanics", only?

Any mass distribution is possible. Mass points, rigid bodies ...

 

[masudr]

Who says "particle mechanics", only?

Any mass distribution is possible. Mass points, rigid bodies ...

We never consider an infinite number of particles. Remember that a rigid body is an infinite number of infinitesimal particles.