Norton's Dome

[ydoaPs]

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

AFAIK, in the entire literature, no one has shown any laws to be broken. It violates Lipschitz continuity, but that isn't a problem. It will be guaranteed that you have a unique solution if your equation is Lipschitz continuous, but failing to be Lipschitz continuous does not entail a failure to have a unique solution. Lipschitz continuity isn't a requirement for Newtonian Mechanics anyway. In fact, this paper argues that there will always be problem cases if you require Lipschitz continuity in Newtonian Mechanics and it is not the case that making such a requirement "can accommodate forcing the particle at rest at the apex to remain there at rest".

 

Now, the question we need to ask is whether the Lipschitz discontinuity is due to the co-ordinate singularity or due to the situation itself. That, I don't know. One could reformulate it in a different co-ordinate system (perhaps with cylindrical co-ordinates and having the force in terms of r), though I'm not sure this has been done to date.

[swansont]

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

 

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.

 

Sorry, realistic? There's nothing realistic about any of this. If realistic is a criterion, then this would not be an issue.

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.

 

Which ball emerges at the end? Because it has to be the one after that.

 

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

 

I think the conclusion is that you apply Newton's laws to the real world, and that you can find mathematical loopholes. the series of collisions is obviously unphysical, since it requires an infinite number of them.

[ydoaPs]

Which ball emerges at the end? Because it has to be the one after that.

Are you talking forward or backward? Forward, none. Backward, the incident ball in the forward version. In the setup, there is a defined first ball in the forward version. So, in the backward version, there is no defined last ball, but there is a defined first ball.

 

Actually, we can add 1 more ball to your infinite collection and have a defined last ball at both ends. If we label the location of the first ball at x=0, then the last ball is at exactly x=2.

[studiot]

First, My apologies.

 

In my post#72 I meant and should have said t = 0.

 

 

 

I realised this last night, just after I shut down ( as one does) along with the thought that this issue and what Swansont proposed earlier was effectively one of Zeno's paradoxes.

 

No object can ever move.

 

The question appears to be how does the motion get started and using distance as the independent variable (as Zeno and Swansont did and I did in error) leads to a Zeno's paradox.

 

The discontinuity is in time not space.

 

ydoaPs, I am not saying your analysis is flawed, (I haven't seen one yet), I am saying Norton's is in the paper linked to, insofar as results are derived that rely on continuity that is not there.

 

Swansont, I did not address my remarks to your set of infinite balls but to the original problem but I suppose the question of initiation is the same.

 

Incidentally there is nothing in Newton's law to suggest that time can or cannot run backwards.

However the nature of a discontinuity is the limits from the left do not equal limits from the right, which is what happens at the instant of commencement of motion.

 

This therefore rather suggests that you can only run time equations backwards in piecewise fashion between points of discontinuity but never include the points themselves.

Edited June 8, 2015 by studiot

[swansont]

 

I realised this last night, just after I shut down ( as one does) along with the thought that this issue and what Swansont proposed earlier was effectively one of Zeno's paradoxes.

 

 

I've been getting a Zeno vibe for a while. It definitely has to do with dealing with infinitesimals, but the details might be different.

 

There's the issue of most systems being defined such that a force acts at t=t₀ and here the force acts at some undefined time after that.

[studiot]

Are you familiar with the particular Zeno paradox I am referring to?

 

Since we are using r as a measure of position,

 

If you plot position v time, the first time derivative, dr/dt (ie the velocity) v time and the second (acceleration) and subsequent derivatives v time, there is a jump discontinuity for each function at t = 0.

 

There is some interesting mathematical discussion of both Norton's Dome and Newton's Laws

 

http://physics.stackexchange.com/questions/39632/nortons-dome-and-its-equation?lq=1

 

and

 

http://physics.stackexchange.com/questions/13557/history-of-interpretation-of-newtons-first-law

Edited June 8, 2015 by studiot

[swansont]

Are you familiar with the particular Zeno paradox I am referring to?

 

Since we are using r as a measure of position,

 

If you plot position v time, the first time derivative, dr/dt (ie the velocity) v time and the second (acceleration) and subsequent derivatives v time, there is a jump discontinuity for each function at t = 0.

 

It's not at t=0, though. At t=0, r=0, v=0 and a=0. If the discontinuity were at t=0 there wouldn't be an issue.

[studiot]

The discontinuity is a t = 0.

 

It is discontinuous at t = 0 because the value of the each function has different limits from the left and limits from the right.

 

So the limit at t = 0 if it exists except by convention can only equal one of them and and convention would have it that the the limit is 0.5{(Lt+) + (Lt-)}, which is neither of these anyway.

[swansont]

The discontinuity is a t = 0.

 

It is discontinuous at t = 0 because the value of the each function has different limits from the left and limits from the right.

 

Same can said of a force that is initiated at t=0, though.

[studiot]

Well yes of course, but the force is the mass times the second derivative of r w.r.t.t. , which we already have in the sequence, just as momentum is the first derivative scaled by the mass.

Edited June 8, 2015 by studiot

[md65536]

I've been thinking about the "infinite series of perfectly elastic collisions" strawman that was introduced...

 

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.

According to the description of the system, at any time there is one and only one mass moving. It's either the first mass or the most recently hit mass. After a time of 2d/v, all of the masses have collided. At that time there is one mass moving, and nothing else to block its path, so that mass will exit the system.

 

I think you would have a tough time proving that all of the masses have stopped moving. You would need to conclude that, before being able to justify any of the claims of physical laws being broken. At best you might prove that there is a contradiction. I don't think you could conclude without contradiction that all of the masses have stopped. Then technically, by the principle of explosion, you can derive that a physical law has been violated, but you could also derive that no physical law has been violated.

 

 

Anyway, that aside, without finding a contradiction yet, I suspect that it might be possible to reasonably speak of the last member of an infinite set without any "the math can't handle it" nonsense. An infinite set is no problem, and ordering it is no problem (this set is well-ordered). There's a well-defined first member, so why not also a last? If we label all of the masses by the order in which they collided, it cannot be that any finitely numbered mass exits the system. But that's no problem because there's an infinite number of masses. It's given that the infinite number of collisions happen in a finite amount of time, the math handles that fine. All of any finite number of masses has a mass that comes after it. Not until counting an infinite number of masses, can you have a mass that has no mass after it. That would be the last mass. I can't derive a contradiction given the description of the system, can anyone else?

 

 

The masses could also be labelled by their position on the x-axis. The last mass would have to have a position of 2d. There is no nearest number to that value, and any n'th mass, where n is a natural number, would have a definite location that is not equal to 2d. But that's no problem because there are an infinite number of masses. The location of the last mass would have to equal the sum of all of the infinite number of spaces between all preceding masses, and the sum n from 0 to infinity of d/(2^n) is 2d (the math can handle it).

---Edit: Then again, that makes no sense. If you count backward from a last mass, you'd need for there to be an infinite number of them before you reached a definite location < 2d, and none of the masses at 2d are in the defined set of masses. So a contradiction seems inevitable. I guess math can't handle physics after all.---

 

 

When someone says "the math can't handle it", without pointing out a flaw in the maths, other than that something doesn't make sense to them, it's usually the case that the flaw is in understanding.

Edited June 11, 2015 by md65536

[studiot]

As a matter of interest since my alignment of point masses and particles caused some eyebrows to be raised.

 

I still contend that this is the normal common or garden interpretation

 

British best practice.

Classical Mechanics : Gregory : Cambridge University press

 

 

A particle is an idealised body that occupies no volume, only a single point in space

and has no internal structure.

 

American best practice.

Analytical Mechanics : Cassidy and Fowles : Saunders

 

 

A concept, involving mass, which we shall have occasion to use throught this text, is that of a particle , or point mass, an entity that possesses amss but no spatial extent.

 

[Robittybob1]

As a matter of interest since my alignment of point masses and particles caused some eyebrows to be raised.

 

I still contend that this is the normal common or garden interpretation

 

British best practice.

Classical Mechanics : Gregory : Cambridge University press

 

 

American best practice.

Analytical Mechanics : Cassidy and Fowles : Saunders

 

 

That would be right for an idealized particle but those particles are never that small that there are an infinite number of physical particles in a finite amount of space.

[md65536]

That would be right for an idealized particle but those particles are never that small that there are an infinite number of physical particles in a finite amount of space.

Still, mathematically it is possible to deal with it---just make each particle half the size of the last and you can fit an infinite number of them in twice the space of the largest particle---it doesn't matter anyway because the masses were defined as point masses, and maths can handle much that is not physically possible. I'm stumped again here, but there's a quote from wikipedia that I think might be required to fully think through this: "As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory."

 

It seems a paradox is certain, but does that require a logical contradiction? If not, does the paradox imply a violation of physical laws in the "infinite collision" strawman? And even if so, is there even a paradox at all in Norton's Dome? And does that also imply a contradiction or violation of a law? Maybe, maybe the evidence was all presented in this thread and I don't understand it, but I think this is all way too much to just hand-wave through.

Edited June 11, 2015 by md65536

[studiot]

That would be right for an idealized particle but those particles are never that small that there are an infinite number of physical particles in a finite amount of space.

 

What particles please bob?

[Robittybob1]

 

What particles please bob?

The ones that are physical particles and also described as "point particles" of course.

For instance mass acts from a point but the size of a particle determines how many you can fit into a certain space (excluding black holes where the dimensions of particles are physically changed).

Edited June 12, 2015 by Robittybob1

[tar]

SwansonT,

 

I think you are right, that the particle has to remain at the top of the dome, and that it cannot move off of its spot, and fulfill the second case unless Newton's first law is violated.

 

But

"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." with our point mass and frictionless surface, that allows ANY small change in the particle being normal to the dome, to cause gravity to instantly pull the particle in the opposite direction of the movement of the dome that caused the tip of the dome to no longer be between the particle and the center of the Earth.

 

So the second case, that of the particle starting to move from its rest position, can not occur, unless the dome moves or the particle moves, and the tip of the dome, the particle and the center of the Earth are no longer lined up. Whether a truck hits the dome, or a Chinaman jumps off a chair on the other side of world and displaces the dome a little, or we have a case of MigL's quantum giggling, the line particle-dome-center of Earth must be disturbed, for the particle to move. With a frictionless surface and a point particle ANY infinitesimal move off of the normal will cause the ball to no longer be at rest.

 

Regards, TAR

Edited June 12, 2015 by tar

[md65536]

There is some interesting mathematical discussion of both Norton's Dome and Newton's Laws

 

http://physics.stackexchange.com/questions/39632/nortons-dome-and-its-equation?lq=1

 

and

 

http://physics.stackexchange.com/questions/13557/history-of-interpretation-of-newtons-first-law

 

Something interesting from the second link, is this:

 

Here is Newton's original statement of the law (Motte's translation):

Law I. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

 

This is different from the modern statement of the law. The wording with "compelled" implies causality, and that the force causes the mass to not remain at rest, whereas the modern statement of the law does not. If you could argue that "Newton's first law" may refer to this originally stated law, then I'd agree that Norton's Dome violates this law. However I think "Newton's first law" refers to what remains a law (ie. the modern version).

 

My whole argument would rest on the differences between the original law and the modern, and on accepting that the changes are necessary ones, that the modern statement fixes flaws in the original, and that the original statement is no longer a valid law.

 

In my opinion, all the arguments that hinge on the "cause" of the mass being knocked off balance, would rely on denying the importance of the changes from the original law to the modern. The part that is being violated is no longer a part of the first law.

 

The modern version from https://en.wikipedia.org/wiki/Newton's_laws_of_motion:

 

First law: When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force.

[studiot]

how is 'acted upon' different from 'compelled' ?

[ydoaPs]

how is 'acted upon' different from 'compelled' ?

The latter could imply some sort of internal property being the causal agent, such as a mental state. And, before you get to it, there actually are people who think that electrons have mental states. It's called panpsychism.

[studiot]

I don't get the connection.

 

'impressed theron' from Newton implies to me the external force in the modern version.

 

With the exception, already noted, that Newton was specific in his use of the plural forces and the modern version only contains one force. and therefore allows the possibility of a contradiction when two or more forces in balanced opposition act on a body, producing zero effect.

Such a contradiction is excluded in Newton's version.

 

But none of this is relevent to md65536's comment.

Edited June 12, 2015 by studiot

[md65536]

how is 'acted upon' different from 'compelled' ?

It is with the full phrasing, "compelled to change by," that it directly states that the force causes the change in state, ie. from "at rest" to not at rest.

"In classical physics, a cause should always precede its effect." https://en.wikipedia.org/wiki/Causality_(physics)

The version with "compelled" (and accompanying details) requires a causal order of events, requiring that the force that compels the body to change (the cause) precedes the change of state (effect).

 

"Acted upon" describes influence or effect, but it does not state that "the body is no longer at rest" is the effect. That is a common-sense assumption. This is difficult to see the difference because in everyday life, the 1st law follows common sense and Newton's first law is causal. But that's not what it says, and for good reason. Perhaps the only counterexamples where the 1st law does not follow common-sense causality are in quantum mechanics, I don't know. "Acted upon" does not imply a causal relationship between the acting force and the change in state. There is no order of events required.

 

I think this is only clear if you intentionally set aside what common sense says and look only at what the science says.

Edited June 12, 2015 by md65536

[studiot]

 

It is "compelled to change by" that states that the force causes the change in state, ie. from "at rest" to not at rest.

"In classical physics, a cause should always precede its effect." https://en.wikipedia...ality_(physics)

The version with "compelled" (and accompanying details) requires a causal order of events, requiring that the force that compels the body to change (the cause) precedes the change of state (effect).

 

Please display the chain of reasonsoning backing up this claim.

 

Newton, of course, knew nothing of Wikipedia, and did not mention causality.

 

I see nothing in the statement to require precedence of the forces over the effect.

 

I repeat and repeat and repeat that there is a difference between the plural and singular in this comparison, because rest is involved.

 

Furthermore The Wiki version definitely implies an effect since it states that there is no effect unless acted on....

Newton backs both horses by stating the positive....'compelled'

 

I have noticed that in several areas of physics carefully wrought statements by truly great men have been changed by lesser men in modern parlance with the result of a problem somewhere.

 

there are two MIT lectures on Ytube where this has happened to Kirchoff in electrical engineering and Lord Thompson in thermodynamics.

 

Edit

 

As a matter of interest Newton and his contempories realised that their mechanics 'fluffed' action at a distance, but they did not know how to resolve the issue (have we truly done so today?) so had to remain content that their strong statements worked at the short ranges they were concerned with. They also lweft some wiggle room and this is cutely discussed by Turner in the Routledge University Physics book Relativity Physics. Turner presents a fascinating rewrite of N1, N2 and N3 to make this compatible with the principle of relativity.

Edited June 12, 2015 by studiot

[StringJunky]

What makes the mass move if it's in equilibrium? The only way I can see it, is that it takes some unknown time to move because there is some physical disequilibrium within the mass, when it sits on the top for some indeterminate duration, which has not yet manifested as movement. Although it is measurably still, it is not settled; in a sense, it's momentum may temporarily be equally distributed in all directions. Is that scientifically sensible?

[md65536]

Furthermore The Wiki version definitely implies an effect since it states that there is no effect unless acted on....

Newton backs both horses by stating the positive....'compelled'

 

I have noticed that in several areas of physics carefully wrought statements by truly great men have been changed by lesser men in modern parlance with the result of a problem somewhere.

I'm not going to get into an argument over whether a physics law from the early 1700s is better than the version accepted today. It doesn't matter, the link in #1 is using a modern version of the law.

 

"Unless" does not indicate what is the effect (and what is the cause).

Consider a cat in a box with a vial of poison that is randomly either broken or not broken at the time it is observed. Consider what is observed:

The cat is alive, unless the vial of poison is broken.

The vial of poison is broken, unless the cat is alive.

 

"Unless" doesn't indicate causality. The cat being alive does not cause the vial to not break. "The cat is alive, unless compelled by a broken vial of poison to change its state to dead" does indicate causality.

 

 

 

 

Edit: Anticipating misunderstanding of my point, this variation should make it clearer:

Consider two cats A and B in a box with a vial of poison that will either randomly be broken or not. Assume the poison is 100% fatal, that if the vial is broken the cats are dead. Assume there is no other cause of death during this experiment. What is observed?

Cat A is alive, unless cat B is dead.

Cat B is alive, unless cat A is dead.

 

"Unless" doesn't indicate causality. Cat A being dead does not cause B to no longer be alive, etc.

"Cat A is alive, unless compelled to change its state from alive to dead by cat B being dead" is NOT something that is observed in this experiment, and is NOT a law at all, while "Cat A is alive, unless cat B is dead," would be a valid law in this experiment if the assumptions were valid.

Edited June 12, 2015 by md65536