A paradox inside Newtonian world

[arildno]

Hurkyl:
Arnold as axiomatized classical mechanics, as far as I know.

 

[Tomaz Kristan]

I think the best way to fix this is to limit to finite numbers of point masses.

This, for one, I think.

another option would be to impose that masses cannot get arbitrarily small.

That also. Both, I think.

Also note this would involve dissipating an infinite amount of energy.

Mass point is already a grave for the infinite amount of energy. Nobody cares, but that's not wise.

People are quite confident in real number modeling of physics, but we have a problem, it will not just go away, if one pretends it's all well. It isn't.

 

[vanesch]

Mass point is already a grave for the infinite amount of energy. Nobody cares, but that's not wise.

Not by itself. Only with a potential that goes in 1/r. It is sufficient to cut off the potential at a small value of r, and you don't have a problem with mass points. There's no problem with mass points and Hooke's force law, for instance.

 

[Tomaz Kristan]

Not by itself. Only with a potential that goes in 1/r. It is sufficient to cut off the potential at a small value of r, and you don't have a problem with mass points. There's no problem with mass points and Hooke's force law, for instance.

You are right. If the mass point is eternal, that is. If you've joined two mass points, the infinite amount of energy was transformed.

 

[vanesch]

Hurkyl:
Arnold as axiomatized classical mechanics, as far as I know.

Yes, but you need a smooth lagrangian function on a finite-dimensional manifold (tangent bundel of the configuration space). As such, you cannot have potentials such as 1/r and you cannot have an infinite number of degrees of freedom.

 

[reilly]

Once again, the problem is not well set. The nature of the interval, half closed or half open makes a difference. And that's a math problem, not a physics problem. After all, physics is defined by physicists, sometimes in desperation over tricky math. In physics, we can test by experiment; without that quality, physics goes into never-never land -- who's to figure.

Also, there's a huge literature going back, at least, to Poincare, Caratheodory -- the outer measure guy --, and ... All very rigorous, and highly mathematical. See also Lanczos, The Variational Principles of Mechanics. All this is applicable to the problem of this thread. I suspect that a Lagrangian approach would require a constraint that all sums be finite, otherwise, it's not clear that the Langrangian makes any sense at all. The forces implied by the Lagrangian multipliers would be interesting indeed.

There is no paradox. And even if there was one, it is hard to see what effect there might be in classical physics. For those who think there is a paradox, what impact does it have on physics?
Regards,
Reilly Atkinson

 

[masudr]

There is no paradox. And even if there was one, it is hard to see what effect there might be in classical physics. For those who think there is a paradox, what impact does it have on physics?

I don't think there is any issue of with classical physics being broken. As far as I know, the issue of contention here is the soundness of Newton's laws.

 

[reilly]

I don't think there is any issue of with classical physics being broken. As far as I know, the issue of contention here is the soundness of Newton's laws.

But Newton's Laws are a substantial part of classical physics -- if Newton is wrong, then classical physics is wrong. So if Newton works in the real world, without any problems -- except in well known exceptions --, then what's the problem if the Laws don't work for some imaginary, nonphysical system?

For physicists, Newton's Laws can become suspect, if not wrong, only if some empirical phenomena, supposed to be governed by Newton, is not governed by Newton. Physics is, after all, an empirical science, and Newton's Laws are, so far, empirically sound.
Regards,
Reilly Atkinson

 

[masudr]

Newton's Laws are rarely placed on a very precise theoretical footing. As a physics student, I've seen Hamiltonian and Lagrangian versions of classical mechanics specified very exactly, but rarely Newton's Laws.

That leads one to suspect that they are not that useful except for elementary problems, the type involving 10 tonne traines on frictionless tracks, sacks slipping down slopes and ladders on walls etc. More sophisticated physics, such as barrels rolling inside barrels rely on other flavours of classical mechanics to reach an easy solution.

Anyway, my point above was that no one in this thread thinks that classical physics has gone wrong. More that Newton's Laws can go wrong if applied naively.

 

[Hurkyl]

So if Newton works in the real world, without any problems -- except in well known exceptions --, then what's the problem if the Laws don't work for some imaginary, nonphysical system?

If Newton's laws are unsound, then they don't work in the real world, because they simultaneously makes every possible prediction -- that we ever got right answers by using them is just a phenomenal streak of luck. (Or, more likely, that we were never using the full force of Newton's laws, but instead using some weaker version which turns out to be sound)

 

[RandallB]

... Newton's laws are unsound, ...
... because they simultaneously makes every possible prediction

What can this possibly mean?

 

[Hurkyl]

What can this possibly mean?

Exactly what it sounds like. If Newton's laws are inconsistent and you wanted to, say, predict the velocity of an object in some situation, then you would find that Newton's laws predict v = 0, and they predict v = 1 m/s, and they predict v = 2 m/s, and they predict v = pi m/s, and they predict v = 34 Joules per meter, et cetera.

But that's inconsistency, not unsoundness; sorry 'bout that. We've been talking about inconsistency this entire thread, so I missed it when masudr switched over to soundness. (And from his post, I assume he meant to say consistency)

 

[reilly]

Newton's Laws are rarely placed on a very precise theoretical footing. As a physics student, I've seen Hamiltonian and Lagrangian versions of classical mechanics specified very exactly, but rarely Newton's Laws.

That leads one to suspect that they are not that useful except for elementary problems, the type involving 10 tonne traines on frictionless tracks, sacks slipping down slopes and ladders on walls etc. More sophisticated physics, such as barrels rolling inside barrels rely on other flavours of classical mechanics to reach an easy solution.

Anyway, my point above was that no one in this thread thinks that classical physics has gone wrong. More that Newton's Laws can go wrong if applied naively.

As both a one-time physics student and as a professor who has taught advanced mechanics numerous times, I suggest your dim view of
Newton's Laws is a bit short sided. All of rotational dynamics, spinning tops, barrels with in barrels can and can be solved with Newton's Laws. Anything can go wrong if applied naively -- my experience as a teacher of this stuff is that many have, initially, a particularly hard time with generalized coordinates in practical problems -- rulers moving to the edge of a table and falling off, for example. And, don't forget relativistic mechanics; don't forget plasma physics, all of which are heavy with Newton's (and Maxwell's and Lorentz's) ideas.

It is also my experience in teaching and in asking questions in oral PhD qualifying exams that students get too excited about the heavy formalism of advanced mechanics, and forget to use simple freshman physics to solve problems like movement of a ladder sliding down a wall... What's elementary?

Are you trying to suggest that somehow the Lagrange equations are not equivalent to Newton's Eq.?Go back in history and check out Whittakers treatise on mechanics, and check out some of the Cambridge Tripos problems, and then claim that somehow Newton is deficient.

And my point is, however one chooses to describe the possible problems with Newton, they are, so far, totally irrelevant to actual physics. Drawing too fine a distinction between inconsistency and unsoundness seems to me to be a semantic distinction of no consequence.

Regards,
Reilly Atkinson

 

[masudr]

reilly:

Please don't read between the lines. I said, "That leads one to suspect that they are not that useful except for..." Never did I suggest that Newton's Laws are deficient in any way.

Are you trying to suggest that somehow the Lagrange equations are not equivalent to Newton's Eq.

No, I'm not. Where did you get that idea from? I specifically said that some problems are easier to solve using different formalisms. Are you trying to suggest that Newton's Laws are superior to other formalisms? Oops! I'm reading between the lines again.

I haven't done every problem possible with ladders on walls, but those that I have done have been exceedingly simple (I know they make unrealistic assumptions etc). Forgive me for thinking they are elementary, but that's exactly my experience of ladders-on-walls type of problems: elementary.

As for doing Cambridge's Tripos problems, I probably would have done them by now had they offered a pure physics course. But since they only offer it with Chemistry or with Maths, I didn't really want to do them. I'll stick to the exams that I am studying for, thank you very much.

 

[masudr]

But that's inconsistency, not unsoundness; sorry 'bout that. We've been talking about inconsistency this entire thread, so I missed it when masudr switched over to soundness. (And from his post, I assume he meant to say consistency)

Yes, I did.

 

[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

$$x_c = \sum_{1 \leq i} \frac{m_i x_i}{m}.$$

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


The "force" on the center of mass is

$$F_c = m \frac{d^2 x_c}{dt^2}$$

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

$$F_c = m \frac{d^2}{dt^2} \sum_{1 \leq i} \frac{m_i x_i}{m} = \frac{d^2}{dt^2} \sum_{1 \leq i} m_i x_i.$$

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

$$\sum_{1 \leq i} m_i \frac{d x_i}{dt}$$

and

$$\sum_{1 \leq i} m_i \frac{d^2 x_i}{dt^2} = \sum_{1 \leq i} F_i$$

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