A paradox inside Newtonian world

[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.

 

[Tomaz Kristan]

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

Wait a minute! A rigid body is a complex of infinite number of rigid bodies. In an ideal Euclidean/Newtonian world, of course.

I thought, everybody knows that.

 

[masudr]

Wait a minute! A rigid body is a complex of infinite number of rigid bodies. In an ideal Euclidean/Newtonian world, of course.

For example,

m=\int_V \rho(x,y,z)\,dx\,dy\,dz

Each infinitesimal volume of the object dx\,dy\,dz at the point (x,y,z) has density given by the function \rho(x,y,z).

In all calculations with rigid bodies, we can usually integrate over the volume and obtain results like the above.

Until you give me an example where I need to consider a rigid body as a complex of infinite number of rigid bodies, in which case I'd happily switch to that definition, I'd rather stick to my simpler, non-circular definition.

I thought, everybody knows that.

I obviously didn't. Why patronise?

 

[Tomaz Kristan]

Each infinitesimal volume of the object dx\,dy\,dz at the point (x,y,z) has density given by the function \rho(x,y,z).

It is just another way of looking at the rigid bodies.

In all calculations with rigid bodies, we can usually integrate over the volume and obtain results like the above.

You can do the same for my (ball) example.

I'd rather stick to my simpler, non-circular definition.

It's not a definition, it's a property of a rigid body.

 

[masudr]

You can do the same for my (ball) example.

I must admit I haven't followed the thread post-by-post. So what is the density function for your ball example?

 

[Tomaz Kristan]

Every ball is 500 times denser than the previus one. And twice smaller by mass.

 

[ObsessiveMathsFreak]

Every ball is 500 times denser than the previus one. And twice smaller by mass.

But does the sum of forces converge?

 

[Tomaz Kristan]

But does the sum of forces converge?

No it doesn't. Why?

But the sum of all forces to any ball - it does!

 

[masudr]

Every ball is 500 times denser than the previus one. And twice smaller by mass.

Where is the "previous" ball located compared to the first ball?

 

[masudr]

But the sum of all forces to any ball - it does!

But it's not any old ball: it's a ball designed so that the sum of forces don't converge, isn't it?

 

[Tomaz Kristan]

Where is the "previous" ball located compared to the first ball?

At 1/10^N.

 

[Tomaz Kristan]

But it's not any old ball: it's a ball designed so that the sum of forces don't converge, isn't it?

Oh, no. They do converge!

 

[Tomaz Kristan]

It is a finite force to every ball. An infinite number of forces sums to a finite number for every ball.

Those numbers go over every finite limit, but all are finite themselves.

 

[ObsessiveMathsFreak]

Oh, no. They do converge!

You just said they didn't!

 

[Tomaz Kristan]

No. Listen carefully!

The sum of all forces acting to every ball is finite. An infinite number of forces adds up to a finite number.

For every ball.

Do you agree?

 

[ObsessiveMathsFreak]

The sum of all forces acting to every ball is finite. An infinite number of forces adds up to a finite number.

For every ball.

Do you agree?

No. An infinite number of forces adds up to infinity. Of course in the limit as the number of forces grows without bound, then sum may be convergent. But we're not talking about such a case. Our scenario here is more akin to Hilbert's hotel than Zeno's paradox.

 

[Tomaz Kristan]

An infinite number of forces adds up to infinity

Not at all.

The sum of all left forces to any ball, is less than twice as the force the of immediate left ball.

Holds for every ball.

Why? Be cause the mass of the immediate left ball is already half of all the others on the left side.

See this now?

 

[ObsessiveMathsFreak]

See this now?

No. The sum of an infinite number of numbers is infinite. Seriously. Look it up.

 

[Tomaz Kristan]

The sum of an infinite number of numbers is infinite.

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

Don't you know that?

 

[ObsessiveMathsFreak]

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

Don't you know that?

\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{2^k} = 1

But you're not talking about a limiting case. You're talking about infinite terms directly.

 

[Tomaz Kristan]

No. Not at all. The sum of forces to EACH ball is a finite one.

 

[ObsessiveMathsFreak]

No. Not at all. The sum of forces to EACH ball is a finite one.

Only in the limiting case. You said you weren't talking about the limiting case. You're talking directly about the case with an infinite amount of particles. The direct sum of an infinite number of numbers is not defined.

 

[Tomaz Kristan]

What's your point OMF?

Do we have a finite force to every ball or we haven't?

 

[ObsessiveMathsFreak]

What's your point OMF?

Do we have a finite force to every ball or we haven't?

No there are an infinite amount of particles.

 

[Tomaz Kristan]

No there are an infinite amount of particles.

So what? You may divide ANY Euclidean body into an infinite number of smaller bodies.

Is it always a problem, or isn't?

 

[ObsessiveMathsFreak]

So what? You may divide ANY Euclidean body into an infinite number of smaller bodies.

Is it always a problem, or isn't?

Not really. You can take the limit of subdivision as the divisions become infinite. From this, one can validly speak of densities, etc per unit volume or whatever.

But to say that anybody consists of an infinite number of smaller bodies is wrong. If that were the case, then the smallest amount of matter would have infinite mass.

 

[Tomaz Kristan]

to say that anybody consists of an infinite number of smaller bodies is wrong.

In the real world yes, it's wrong. NOT so in the Newtonian abstract world.

If that were the case, then the smallest amount of matter would have infinite mass.

This is nonsense.

 

[masudr]

I think there is confusion between infinitesimal volumes and actual particles. Do you think they are the same thing, Tomaz Kristan?

 

[Tomaz Kristan]

I think there is confusion between infinitesimal volumes and actual particles.

I don't do infinitesimal volumes. I do only finite volumes in my case. For each real epsilon, there are only a finite number of grater than epsilon volumes.

But each volume is a finite one, non the less.

For the half, quarter, 1/8 and so on parts of any Euclidean body (like cube) is the same. Only those do not grow by density. But they may be squeezed to my balls and do just that.

In the abstract Newtonian realm.

 

[Amr Morsi]

Hi Tomaz / All,

Thanks to hear for the following, maybe it will end up the long debate.

Let's not get into deep mathematical discussions. The main debated problem here is the physical rule. By the way, no scientist until these days said that Newtonian mechanics are wrong in structure. But, they agreed upon it is non relativistic in addition of being classical.

The very first Rule is only for rigid bodies. What we are talking about here are bodies of non-constant vector lengths (displacement).

Besides, this is not the body of the Newtonian Mechanics. But,of course, it is important.

The rule says that a rigid body in the vacuum can be having an angular velocity (directional acceleration) around his centre of gravity, which may be moving with constant velocity. This can be easily deduced by the general laws of motions:
1. Forces' Resultant to applied to the centre of "mass" to get acceleration of the centre of "mass".
2. Torques' Resultant to be applied to the body to get angular acceleration.

Hint: This is applicable in the 3-D Motion (and also to Relativistic Mechanics after some simple modifications).

Amr Morsi.

 

[reilly]

Happy New Years. The situation this year is just as it was last year: there is no paradox. the problem is not well set, as mathematicians say, and thus has no solution. Let's look again.

First, let's look at the limit of the system as N - > infinity, after the equations of motion have been solved. That is, first solve for the system's behavior with a finite number of masses. Then, of course, there will be no net force acting on the CM. So, the sequence of forces acting on the CM is 0,0,0,...0,...
Under most circumstances, this sequence converges to, guess what, 0.

Now, let's take the limit of N-> infinity, before solving the equations of motion. If I've read things correctly, the gravitational potential of the system at any point r, along the line, will be V = - G*SUM(over n){ (1/2)^n/|r - (1/10)^n|}.

It's a fair bet to claim that this series does not converge. So, we get a different answer, non-answer in fact, than above.

QED -- the problem has no solution. (Even if one is clever enough to note that the potential has poles for r= (1/10)^n, one can finesse such poles as is usually done in most 1/r potential problems. See most any text on Maxwell+Newton+Einstein, Jackson, for example.)

In this case, one road leads to Berkeley, another leads to Palo Alto, another leads to Cambridge,MA, and so on. "Can't get there from here."

No paradox, just a badly stated problem, one that makes no sense. Sorry 'bout that.
Regards,
Reilly Atkinson

 

[Hurkyl]

V = - G*SUM(over n){ (1/2)^n/|r - (1/10)^n|}.

It's a fair bet to claim that this series does not converge. So, we get a different answer, non-answer in fact, than above.

Diverges at r=0, and is undefined whenever r is the coordinate of one of the mass points. Converges everywhere else.

 

[Tomaz Kristan]

Converges everywhere else.

And above all, converges for the system as stated in post #1.

 

[Hurkyl]

No, it diverges at r=0 and is undefined when r is the coordinate of the mass points.

 

[reilly]

Diverges at r=0, and is undefined whenever r is the coordinate of one of the mass points. Converges everywhere else.

Absolutely correct. But that's sufficient to make my point.
Regards,
Reilly Atkinson

 

[Tomaz Kristan]

The sum of forces to every ball is finite and very well defined.

And you know that, I guess.

 

[Tomaz Kristan]

Let take the rightmost ball! It had been all the left balls condensed into into the second rightmost ball, the force between those two would be greater than it is now. And quite finite, still. Sure.

See? See the implication to every other ball?

 

[ObsessiveMathsFreak]

The force on the center of mass is zero for every configuration of N masses. In the limit as N-> infinity, the force is still zero. With an infinite number of particles, the force is undefined. We've been over this before.

You keep claiming that teh center of gravity moves to the left, but you haven't proved that. If it does move the the left, how fast does it go? What is the value, in Newtons, of the force on the center of gravity. Please don't say something like:

Every ball has a finite force to the left. There are an infinite number of balls. No leftmost ball.

Then the force is positive to the left.

Do you agree?

Just try and come up with an actual figure. Some number that represents how much force is on the center of mass. You claim it will accellerate. If so, by how much? If you can't give this answer, then the problem is not well posed.

 

[Tomaz Kristan]

I don't care what happens later. All the forces in one direction is already an illegal situation. Remember the third law of sir Isaac Newton?

 

[my_wan]

If this -theoretical paradox- holds it would allow a propulsion device in a device similar to the one about to be described. It wasn't thought of by myself and I,ve never seen an answer. I haven't ever seriously considered it because of the source I have now forgotten. Originally I thought the containment parameters were impossible but as I am now considering it apparently it is not.

Containment surface (hole)
Cut a hole in the center of a flat sheet of metal with the following shape.
Start at position x=0, y=1 as the center point to draw a partial circular line through the lower two quadrants beginning at x=-5 through the -y axis and ending at x=5. Define L as the distance from x=-5 to x=5. Now for each point on the partial circular line using the straight line formula of length L define a corresponding point in the upper two quadrants where the straight line always passes through point x=0, y=0.

The beginning parameters x=1, center of partial circle, as well as x=-5 and x=5 or length can be changed to taste. More efficient shapes are definable that reduces torque or acceleration when the rod passes the x axis.

Device
Now cut a rod of length L that can spin inside the container. Mount a motor with the shaft at position x=0, y=0. Cut a slot in the rod such that the motor will spin the rod but allow the rod to slid back and forth, and mount it within the container. Add a control devise to the motor such that the RPM remains constant under changing torque.

If the paradox holds and you spin this device clockwise an excess of MV will always exist in the direction of the +y axis offset in the +x direction due to the forced containment accelerations. Using the device in pairs with opposite spin would be necessary to balance rotational torque. Proof of concept could be done with one driving a toy car up an incline.

Theoretical implications
It has occurred to me that even if it works to whatever degree it doesn't have to violate the second law.
Law - No motion or system of motions may change the total momentum of an (enclosed) system.
The question then is; 'Is this an enclosed system?'
Just as a pure thought experiment the answer is no. First we are providing energy to maintain a constant RPM. Secondly this system is by definition radiating gravitational radiation. Could that be why it doesn't work?
This is worth figuring out even for purely theoretical reasons.

 

[ObsessiveMathsFreak]

I don't care what happens later.

Considering that what happens later determines whether you are right or wrong, I would have thought otherwise.

All the forces in one direction is already an illegal situation. Remember the third law of sir Isaac Newton?

Unless of course, all the forces happen to sum to zero. Or infinity. If you could just check that, that would be great.