A paradox inside Newtonian world

[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?

 

[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.