A paradox inside Newtonian world

Hurkyl

Newton's laws are rather informal, and certainly not complete. e.g. you've listed nothing that tells you, e.g., that a particle is something with mass and position. (which is a problem, because I don't think you can even state Newton's laws until you've postulated that particles have mass and position)

Before asking this question, I did a brief search for axiomatic Newtonian mechanics, and found Axiomatic foundations of Classical Particle Mechanics. Some key points about this paper are:

The first axiom is that there are only finitely many particles.

In theorem 3 (which deals with center of mass), they remark that the assumption of finitely many particles is essential to their formalism.

If we used their formalism, then there is no paradox: your construction is illegal.

If we took their formalism and threw out the axiom that says there are finitely many particles, then you don't have a center of mass theorem, and once again there is no paradox.

Tomaz Kristan

Page 258 bottom and 259 top:

But an essential generalization of our axiom system would be obtained if we were to replace P1 by the axiom: "P is nonempty, and either finite or countably infinite." If the axiom P1 were to be liberalized in this way, however, then it would probably be desirable to add some additional axioms, as to insure that the total mass and kinetic energy of the system be finite.

Hurkyl

I repeat::

Tomaz Kristan

No. The mass center can be very well defined for the infinite number of particles, when the mass remains finite. At least.

Tomaz Kristan

The paradox does not live only with the infinite set of bodies. You may consider those left balls as glued together, and the Jupiter as the second body on the right side.

The paradox blooms fine, with "only" the infinite divisibility of the matter.

vanesch

Yes, but so is continuum mechanics then...

That's the "problem": we use Newtonian mechanics regularly with an infinite amount of "mass points".

Of course, Newtonian mechanics, limited to two mass points, and non-zero total angular momentum, is an entirely consistent axiomatic system. With one mass point also

Hurkyl

Right; that's why that paper is an axiomatic foundation for particle mechanics, and not for continuum mechanics. I, actually, would really like to see a more general axiomatic foundation, it's just that this was all I could find.

Actually, if you use the techniques of nonstandard analysis, then these axioms are adequate. (you only need hyperfinitely many particles to approximate your continuum)

e.g. In Tomaz's original scenario NSA tells us that a particle of infinitessimal mass gets flung rightwards at transfinite speed, and that exactly makes up for the missing momentum. (That's why I was making a big deal about the behavior about the origin, because that's my best guess as to the standard analog)

Tomaz Kristan

Every ball (no matter how small) has a much bigger left drag, than the right drag. It quickly escapes from the gravity of its right neighbor.

No right moving whatsoever!

Hurkyl

In the nonstandard model we'd use to analyze your scenario, there's a leftmost ball.

Tomaz Kristan

There is NO leftmost ball at all.

ObsessiveMathsFreak

But there is in the limiting case as the number of particles goes to infinity.

Tomaz Kristan

Limit for what? For the "leftmost" ball speed after a second? For the force between the two "leftmost" balls?

Nothing like that exists.

ObsessiveMathsFreak

In the limit as the number of balls goes towards infinity. As we logically increase the number of balls, in each of our cases, there is a leftmost ball.

Tomaz Kristan

No, it doesn't go that way and you know that.

Hurkyl

There is, in the nonstanard model. It contains H balls, where H is a transfinite (hyper)integer. The H-th ball is the leftmost.

radou

This discussion seems to be an ideal example of an infinite process.

Gelsamel Epsilon

What does transfinite mean? Boundless but not infinite?