Norton's Dome

[studiot]

 

You present a poor argument. You make a case but don't give us your evidence for the things you argue.

 

 

Funny I thought my post#43 quoted exactly Newtons words (at least the English version).

 

And, as I pointed out, they differ significantly from yours.

 

 

As an example suppose I said I was going to give you some £s, and handed you one £.

 

It really is also instructive to consider the inverted pendulum theorem I mentioned in post#41.

 

 

 

You present a poor argument.

 

 

In particular I am not the one who started this thread with a link to a professor of history, presenting this subject such that his opening equation is flawed as it does not pass a simple high school dimensional analysis test.

 

 

(1) d²r/dt² = r^1/2

Edited June 6, 2015 by studiot

[Robittybob1]

Why would the inverted pendulum theory have anything to do with this topic? I have never heard of it.

[swansont]

I agree you can be at rest but any random influence can disturb that equilibrium. They seem to call that "spontaneous" maybe you need to argue whether random and spontaneous are synonymous?

are random experimental errors influencing the outcome "spontaneous"?

 

There are none in this thought experiment. It is an idealized case.

But the mass in Norton's dome is being affected by (at least) two external forces.

 

Which cancel at r=0, yielding zero net force. Meaning the body must persevere in its state of rest.

The issue is... for how long, or until when, does "remains" apply? By a strict reading of the law, it applies "unless" a force acts upon it. The "remains" requirement disappears if the "unless" is satisfied. It does not have to last.

 

It lasts until a force is present.

[md65536]

The mass is at r=0 and there is no force on it, so it must remain at rest. That's what Newton's first law says. Is that false?

Do you agree that this is a fair restatement of yours, that says the same thing but is true in every interpretation?:

 

"The mass is at r=0 and there is no force on it. Newton's law says it must therefore remain at rest (unless acted on by a force)."

 

1. This is true.

2. The "(unless acted on by a force)" can be ignored because it is already given.

3. The ignored part can only be ignored as long as there is no force.

4. Half of the law doesn't remain in effect longer than the full law. It isn't true that "the mass must remain at rest" if there is a force acting on the mass.

5. If the mass is not at r=0, there is a net force on the mass.

6. Since the mass is initially at rest and must remain at rest, if the mass is not at r=0 then there is a violation of the first half of the law.

7. The first half of the law (the mass must remain at rest) is only violated when the mass is not at r=0, in which case there is a net force and the full statement of law is technically satisfied.

8. If only half of the law is violated, the law is not violated.

 

Do you disagree with any of these statements?

 

 

I believe there is a trick in your statement regarding point 4. You're ignoring half of Newton's first law because it's redundant, and expecting that the first half must "remain" true even when the ignored second half no longer requires it.

Edited June 7, 2015 by md65536

[Robittybob1]

 

There are none in this thought experiment. It is an idealized case.

 

Therefore why do you call it "a dome shape and the top is a point of unstable equilibrium", if the particle will remain there at R=0?

[md65536]

Therefore why do you call it "a dome shape and the top is a point of unstable equilibrium", if the particle will remain there at R=0?

Because it fits the definition of unstable equilibrium. "If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away." http://en.wikipedia.org/wiki/Mechanical_equilibrium

 

Remaining in equilibrium for an arbitrarily long time is consistent with the first law.

[studiot]

I find it disappointing that this thread has fallen short of the usual high standards at ScienceForums.

 

This is on two main counts.

 

Firstly:

I do not expect to have to go trawling the net to unearth the main thrust of the issue for myself, I expect that presented in the opening post.

 

Secondly:

The issue is essentially one of the mathematical analysis of the situation and no mathematics has been presented. When posted the originator's first equation it was ignored.

 

Now both of these counts are frequently used to rebut wooly posts so why is this thread any different?

 

In point of fact there are several possible ways to continue Norton's analysis mathematically.

Since there are inconsistencies inherent in his mathematics, whichever way one proceeds, we need to have at least one approach presented for proper (mathematical) discussion.

[swansont]

8. If only half of the law is violated, the law is not violated.

 

Do you disagree with any of these statements?

That one.

 

I believe there is a trick in your statement regarding point 4. You're ignoring half of Newton's first law because it's redundant, and expecting that the first half must "remain" true even when the ignored second half no longer requires it.

I don't belive any of it is redundant, and I'm not ignoring half of the law.

 

I want to know what the force is that initiates the motion. The force that is necessary, under the first law, such that the object not remain at rest.

[md65536]

Okay excellent, I feel like we're making progress instead of just repeating mutually consistent statements. I feel like we're near a pinpoint of where we disagree.

That one.

I disagree with you because "An object at rest must remain at rest" is not a law. Even if the situation requires that the statement must be true, there is no law that demands it. The applicable law only requires that "An object at rest must remain at rest unless no external force is acting on it."

In the thought experiment the half-law remains true until some unexplained event (that is not a new force) makes it not true. In reality such an event may be possible (quantum randomness or whatever) or not (perhaps there is always a force that knocks things off balance). But it doesn't matter because even if the thought experiment is describing an impossible event, the event doesn't violate the full first law.

I want to know what the force is that initiates the motion. The force that is necessary, under the first law, such that the object not remain at rest.

There is a force everywhere on the dome that pushes the mass away from r=0, except at r=0. That is the force that satisfies the first law.
Only while the mass is at r=0, is there no force, and Newton's law says it must remain at rest. And it does! For the entire time that the mass is balanced, Newton's law is followed.
If the mass is not at r=0, there is a force, and Newton's law is satisfied (even if a common sense understanding of it is not).

The law doesn't say anything about "initiating" motion. The law does not say that the external force that allows that the object does not remain at rest is one that does any initiating. The law allows the case where the "not being at rest" comes at the same time (or perhaps even before) the presence of the external force. A lack of initiation does not break the law. It violates common sense.

It would be a mistake to have the law conform to common sense, and in this case I don't think your interpretation of the law is consistent with reality. I think that your interpretation of the law is too easily violated in nature at the quantum scale, and so it could not remain a law.




Newton's law is not equivalent to "An external force causes an object to no longer remain at rest," any more than "An object not remaining at rest causes there to be an external force." Newton's law is time-reversible, and this is not. (probably wrong)

Edited June 7, 2015 by md65536

[swansont]

I disagree with you because "An object at rest must remain at rest" is not a law.

That's not a valid reason for disagreeing, because I'm not citing that.

 

 

Even if the situation requires that the statement must be true, there is no law that demands it. The applicable law only requires that "An object at rest must remain at rest unless no external force is acting on it."

Yes. What is the force that causes the object to not be at rest?

 

In the thought experiment the half-law remains true until some unexplained event (that is not a new force) makes it not true. In reality such an event may be possible (quantum randomness or whatever) or not (perhaps there is always a force that knocks things off balance). But it doesn't matter because even if the thought experiment is describing an impossible event, the event doesn't violate the full first law.

What is the force that causes the object to not be at rest?

 

There is a force everywhere on the dome that pushes the mass away from r=0, except at r=0. That is the force that satisfies the first law.

Only while the mass is at r=0, is there no force, and Newton's law says it must remain at rest. And it does! For the entire time that the mass is balanced, Newton's law is followed.

If the mass is not at r=0, there is a force, and Newton's law is satisfied (even if a common sense understanding of it is not).

The law doesn't say anything about "initiating" motion. The law does not say that the external force that allows that the object does not remain at rest is one that does any initiating. The law allows the case where the "not being at rest" comes at the same time (or perhaps even before) the presence of the external force. A lack of initiation does not break the law. It violates common sense.

 

It would be a mistake to have the law conform to common sense, and in this case I don't think your interpretation of the law is consistent with reality. I think that your interpretation of the law is too easily violated in nature at the quantum scale, and so it could not remain a law.

 

Newton's law is not equivalent to "An external force causes an object to no longer remain at rest." Newton's law is time-reversible, and this is not.

Really? Newton's law is time reversible? So no thought experiment could violate this?

 

Consider this scenario: You have point masses. The first two are separated by d, the next one at d/2, the next at d/4, etc. An identical mass is incident upon the first, traveling at speed v, and initiates a series of perfectly elastic head-on collisions, i.e. in each collision the incident mass comes to rest and the target proceeds with momentum mv. The length of this system is 2d, so the collisions take a finite amount of time (2d/v)

 

The problem is, no ball ever exits the system. There are an infinite number of balls, so there is no "last ball" that can exit (just like there is no point "next to" r=0). So if we reverse this, starting at time t= 2d/v, we wait until the clock counts down to zero and then poof, a ball emerges. Conservation of momentum, conservation of energy, and time reversal all fail.

 

So I think you can't hang your hat on time reversal above.

[michel123456]

I want to know what the force is that initiates the motion. The force that is necessary, under the first law, such that the object not remain at rest.

The mass is under gravitation and it does not move because the action-reaction vectors are exactly perpendicular to the surface of the dome. A simple deviation is sufficient to make the mass move. But anyway IMHO you are correct, even in this case the law is not violated because there is a force and we know where it comes from.

I find it very peculiar that some people agree with the idea that after some large amount of time things change without a cause, as if time itself was the cause. In fact, time is not a cause of anything. Time is not one of the 4 interactions.

Edited June 7, 2015 by michel123456

[swansont]

The mass is under gravitation and it does not move because the action-reaction vectors are exactly perpendicular to the surface of the dome.

Action-reaction is not in play here. Saying the net force is zero is not an issue of action-reaction forces.

[michel123456]

Action-reaction is not in play here. Saying the net force is zero is not an issue of action-reaction forces.

??

the term "apex" of the dome means that this point is where the surface tangent is perpendicular to the gravity vector. If gravity is not perpendicular to the tangent where the mass is, there is a vector decomposition and the mass moves.

[md65536]

That's not a valid reason for disagreeing, because I'm not citing that.

Alright, let's drag this on.

You disagreed with the statement "If only half of the law is violated, the law is not violated."

You mean that if "A mass at rest must remain at rest" (only this half) is violated, then even if "the mass is acted on by an external force" (this half is not violated), then "A mass at rest must remain at rest unless the mass is acted on by an external force" is violated.

If this is not what you mean, then what do you mean?

 

 

What is the force that causes the object to not be at rest?

The force that is present at r>0. But I would not use the word "causes". The cause is intentionally unspecified in the thought experiment.

 

 

Really? Newton's law is time reversible?

Yes. Your link in post #1 says so. http://www.pitt.edu/~jdnorton/Goodies/Dome/

 

 

Conservation of momentum, conservation of energy, and time reversal all fail.

Which of those is Newton's first law? Edited June 7, 2015 by md65536

[swansont]

Alright, let's drag this on.

You disagreed with the statement "If only half of the law is violated, the law is not violated."

You mean that if "A mass at rest must remain at rest" (only this half) is violated, then even if "the mass is acted on by an external force" (this half is not violated), then "A mass at rest must remain at rest unless the mass is acted on by an external force" is violated.

If this is not what you mean, then what do you mean?

I thought by half a law you meant the "at rest" vs "uniform motion"

 

The rest cannot be divided.

 

The force that is present at r>0. But I would not use the word "causes". The cause is intentionally unspecified in the thought experiment.

The cause cannot be identified in the thought experiment. For good reason — there isn't one.

 

 

Yes. Your link in post #1 says so. http://www.pitt.edu/~jdnorton/Goodies/Dome/

It was rhetorical.

 

Which of those is Newton's first law?

You didn't say first law. Anyway, there's no motion after t=2d/v, and there's motion beforehand, with no external force. So the first law is violated.

[studiot]

Is there any good reason for not discussing the mathematics of the situation?

 

 

 

What is the equation of motion of the point mass on the dome?

 

Can it be proved that this equation satisfies the Lipschitz conditions?

Edited June 7, 2015 by studiot

[swansont]

Is there any good reason for not discussing the mathematics of the situation?

 

What is the equation of motion of the point mass on the dome?

 

That's the second law. It doesn't violate the second law, AFAICT, since the solution comes from the second law.

[studiot]

 

That's the second law. It doesn't violate the second law, AFAICT, since the solution comes from the second law.

 

 

What doesn't violate N2, that you seem to imply violates N1?

[ydoaPs]

The math can't handle the motion from r=0 to the adjacent point, because mathematically there is no adjacent point. Even though calculus deals with infinitesimals, it doesn't deal with that.

There's NEVER an adjacent point. Between any two points lies another.

[Delta1212]

There's NEVER an adjacent point. Between any two points lies another.

I think that was what he was implying.

[ydoaPs]

What doesn't violate N2, that you seem to imply violates N1?

Yeah, that's my quandry about violation of the first law. If it satisfies the second, I'm not sure how it could fail to satisfy the first. If at every instance in an interval of time F=ma holds, there is no subinterval such that there is acceleration without a force.

I think that was what he was implying.

If that's what he was implying, then he was implying that physics can't handle motion at all.

[studiot]

 

There's NEVER an adjacent point. Between any two points lies another.

 

 

What I think swans means is that the first and second time derivative of r are both discontinuous at r = 0.

Edited June 7, 2015 by studiot

[ydoaPs]

Consider this scenario: You have point masses. The first two are separated by d, the next one at d/2, the next at d/4, etc. An identical mass is incident upon the first, traveling at speed v, and initiates a series of perfectly elastic head-on collisions, i.e. in each collision the incident mass comes to rest and the target proceeds with momentum mv. The length of this system is 2d, so the collisions take a finite amount of time (2d/v)

 

The problem is, no ball ever exits the system. There are an infinite number of balls, so there is no "last ball" that can exit (just like there is no point "next to" r=0). So if we reverse this, starting at time t= 2d/v, we wait until the clock counts down to zero and then poof, a ball emerges. Conservation of momentum, conservation of energy, and time reversal all fail.

 

So I think you can't hang your hat on time reversal above.

emphasis mine

 

I'm not sure that's true. There's the same amount of balls the entire time. There's the same number of balls moving at any one time. Since there's the same amount of balls and the masses of the balls are constant, the mass is the same throughout the interval.

 

Where is the violation here? It seems to me that it is just a violation of common sense relying on common sense's known trouble handling infinity. Yeah, a ball emerges from the end, but there was always a ball moving. There's no point in the interval at which there is no motion which you can use as a starting point in the time reversal to show momentum showing up from nowhere.

[md65536]

Anyway, there's no motion after t=2d/v, and there's motion beforehand, with no external force. So the first law is violated.

This probably needs its own thread, but the conclusion is related, so... I disagree.

 

Consider this scenario: You have point masses. The first two are separated by d, the next one at d/2, the next at d/4, etc. An identical mass is incident upon the first, traveling at speed v, and initiates a series of perfectly elastic head-on collisions, i.e. in each collision the incident mass comes to rest and the target proceeds with momentum mv. The length of this system is 2d, so the collisions take a finite amount of time (2d/v)

 

The problem is, no ball ever exits the system. There are an infinite number of balls, so there is no "last ball" that can exit (just like there is no point "next to" r=0). So if we reverse this, starting at time t= 2d/v, we wait until the clock counts down to zero and then poof, a ball emerges. Conservation of momentum, conservation of energy, and time reversal all fail.

If you have an infinite number of point masses arranged as such, then you have infinite mass with infinite density in the neighborhood of 2d. To be realistic, you could not assume Euclidean geometry or universal time. Realistically, you'd have a black hole which would absorb the momentum and move infinitesimally slowly toward 2d. From an outside perspective, the process would not have to complete in a local time of 2d/v, it could go on forever (local time). There's no violation of Newton's 1st law. But suppose we assume Newtonian physics still applies, admitting that the situation is not physically real. The neighborhood of infinite mass could still move infinitesimally slowly toward 2d, and conservation of momentum is not violated. Or a single mass could emerge from the system, it doesn't matter which one, and conservation of momentum is not violated. But suppose we assume that the impossible description of the system is still correct and neither of those cases happens. Then yes, you can have any law you want violated by specifying the system so that it is violated with some impossibility.

 

If no ball exited, then the reverse direction needn't violate any laws either because the infinite mass can have finite momentum with infinitesimal velocity.

 

The difference between that and Norton's Dome, is that the dome is a conceivably physically possible reality, and the dome isn't specified in a way that a law is required to be broken (though I guess that's not agreed upon).

Edited June 7, 2015 by md65536

[studiot]

 

and the dome isn't specified in a way that a law is required to be broken (though I guess that's not agreed upon).

 

 

Despite 74 posts in this thread no one has shown any laws to be broken.