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.