@@Alice_Sweicrowe the external forces are balanced in equilibrium when the object is at rest: their vector sum is 0. Jade shows this with g = - N in a diagram.
The issue I have with the suggested time reversal or time symmetry proof is that time symmetry is theoretical and that the slope and final location of the ball will only obey that if the top of the dome is sufficiently thin, and it's difficult to know how many atoms can make it sufficiently thick. Besides we need to work at the molecular level because any dome woulds must be stable
In the mathematical solution, the dependence on t^4 presents a discontinuity in the 5th derivative of position, meaning that the "acceleration" of the force suddenly changes from 0 to non 0 at the excitation time. This shouldn't make physical sense, in the frame of Newtonian mechanics, because the first law of motion should trace back to all possible causes of motion. Not only should a body experience zero acceleration in the absence of a net external force, but it should also experience zero jerk (change in acceleration) in the absence of a change in the net external force, and so on. Here, we're actually introducing an instantaneous "force" equivalent to the 6th derivative of position, which causes a discontinuity in the 5th derivative, and propagates all the way to the position, appearing to start moving spontaneously. Therefore, if we restrain Newton's first law to just encompass acceleration, it's easy to destroy determinism, just design a system where acceleration is 0 but its derivatives non 0 (such as the dome). Then you forget what's happening beyond that point. The discontinuity in the derivatives of the acceleration isn't given by the system itself, but it behaves as another initial condition, the one that actually determines the excitation time. It's like saying that you don't know the origin of the changes in the force, and the changes in the changes of the force, then, of course, the system is indeterministic, you are simply lacking information. But if we describe the system as just the dome, there should not be any such discontinuity, thus, no excitation. Instead, if we include all its derivatives, we must determine the behavior of all forces and changes to these, and the ball will move accordingly, therefore determinism is conserved.
All of this takes a funny shade when considering it from a game dev perspective. Gives a good framework for shaping the start and stop times of something's motion, for example.. sentient motion operates on that 6th derivative (or higher) level, it's not just raw changes to position, velocity, or even acceleration. It's changes in rates of changes of rates of changes, ad nauseum.
Hmm. Former theoretical physicist here, and this is the first time I've seen this... but my read (after reading his paper too) is that this is a convincing argument that Newtonian mechanics (and really also Lagrangian or Hamiltonian dynamics, you'd have the same problem) doesn't work in non-Lipschitz-continuous situations. Given the number of other places where it fails (relativistic, QM, etc) that isn't fundamentally shocking, it's more a "good to know - if you're not Lipschitz, you'll have to go beyond any simplified world picture" note. Fascinating. Also, I like how his example includes spontaneous symmetry breaking, as the descent angle is completely arbitrary. It highlights how once you lose Lipschitz, all determinism goes out the window!
Indeed, this is our first ODE course if calculus of variation techniques cannot help refine the conclusion (surprisingly sometimes it can): Banach leads to both uniqueness and existence, Schauder only leads to existence. That is also where the "paradox" of breaking of "conservation of momentum" comes in for Brownian motion when we have infinite p-variation when p < 2 (meanwhile one can somehow save determinism by adding more information, e.g. description of higher order processes)
Not a physicist here. Does this article prove that there is the excitation time after which the ball will move? To me it looks like two different scenarios combined to create a false sense of continuity. We can just as well consider a scenario of a ball standing on a flat surface, and after the time T rolling with uniform motion from exactly the same position.
@@YonatanZunger sure but isnt it a relevant question whether (this sort of) non-Lipschitz situation can ever have physical existence? A perfect dome seems impossible to make, and it seems just as impossible to balance a ball on its point
Applying the "exact energy" to flick the ball up to the dome apex results in the ball taking infinite time to reach that apex. When you try to run this in reverse and realize that the ball can indeterministically choose any path, you have to start at and work backward from T=∞, which you can't do.
Exactly. In the described scenario the ball is always moving to the apex, never reaching it. The only way to get the ball at the apex at an X time with horizontal speed = 0 is to place it there.
This would be true of any dome which had a finite radius of curvature at its apex. However the whole point of Norton’s dome is that its radius of curvature is zero at the apex, meaning that a ball rolling up can reach the top in finite time.
@@mikebrown5223 the problem is not with the curvature radius. It's about limits. At the apex the ball has no force being applied to it and no velocity. However, how does something that is moving stops without a force being applied to it? It can't. The ball near the apex of the dome is subjected to a force and a speed very very close to zero, but it will never reach exactly zero (only at infinity).
@@mikebrown5223 not without friction, because when it reaches the zero curvature area of the apex with a non-zero velocity, it would continue rolling to the other side with nothing to stop it, no?
And if it doesn't exist at all it could be either Trump's brain, not to confuse with a Boltzmann brain, this entity COULD actually exist, or a "metaphysics" course sense in a Spanish speaking university
@@achillesglacia7700 Current academic physics does care a lot about abusing mathematics for it's weird dogmatic reasons. Wolfram's new paradigm is a different case, and there are also other constructivist pure mathematicians who contribute also to physics, Louis H Kauffman, Norman Wildberger etc.
Mathematics, both applied and pure, are not a scientific field@@ThatTimeTheThingHappened. Mathematics is more closely related to a blend of language and music.
In the dome paradox, the "last instant of rest" is like the supremum of a set: it marks the boundary between rest and motion. However, there is no "first instant of motion" because motion begins in an open interval where any candidate for the first moment can always be preceded by an earlier one. This is similar to how a set like {x > 0} has no minimum, even though it approaches zero. Thus, the transition from rest to motion lacks a specific starting point.
Perhaps this all condenses down to quantum mechanics whereby the ball and dome are not truly completely at rest with respect to one another (and to the rest of the universe), but all it would take is some random effect to start the ball moving discernibly.
@@mauricegold9377 what you’re saying is true, but outside the scope of Newtonian physics. Otherwise just by having heat, the ball would move enough to move its centre of gravity out of equilibrium. Newtonian physics is a simplification of all of these things which is why everything is a ball with a net force.
I agree... In my opinion, continuity (and therefore its siblings like uncountable infinities) simply messes with our logic and intuition. This example of "there is no first moment of motion" is yet another delightful example of these alien realities we only discovered and tamed thanks to mathematics. I think our ancient philosopher friend Zeno would have hated this ^^
To the people claiming that non-smoothness or inexistence of snap at t = T makes the example unphysical, while I don't see Newton's laws prohibiting it, I will instead change the example to a smooth one: Note that in the paper r is distance along the surface of the dome, not just the horizontal distance, thus v = dr/dt. h is the loss of height so it will be positive. Use r(t) = e^(-1/(t-T)) for t > T and r(t) = 0 for f 0 and h(0) = 0. This follows from mgh = 1/2mv^2.
I feel like it should be more clear that this argument is not saying balls can spontaneously start rolling. It's saying that Newton's laws fail to describe some edge cases. We know that the ball will not start rolling, but we cannot disprove it with Newton's laws. This is just an issue with the framework. Let me reword this due to comments: we know that within Newton's framework, balls are not supposed to spontaneously roll. What's being argued here is that under the same framework, balls can technically start rolling spontaneously in certain edge cases. If that's actually true, there is a flaw in the framework. This really has nothing to do with the real physical world. Reword #2: it's pretty clear Newton's model was intended to give a deterministic prediction given an initial state. If it fails to do so in some situations then the model is flawed. The model is not attempting to account for any real world phenomenon that might cause an indeterminate solution.
The physical realm is indeterministic. The maths do not figure into their equations the physical aspects of matter due to its varying energetic states and pliant tendencies.
I think you haven't understood the point of 16:10-16:35. Your statement "We know that the ball will not start rolling" doesn't make sense - it sounds like you're trying to describe what would happen if we set up the system perfectly in the real world, but it is not possible to set up the system perfectly in the real world (because, among other things, physical matter is not an actual continuum).
@@DrJulianNewmansChannel In that case is it really a problem with the theory? Newton's laws describe how things behave in reality, they aren't prescriptive, they are descriptive. Should we be surprised that they fail to describe situations that are physically impossible? The laws describe reality, not prescribe how things should behave in a non realistic scenario.
I'll understand the 'argument' better when someone explains how she suddenly pastes the term "excitation time" into her spiel without definition and we don't see it again until some 9 minutes later: "...or as Norton calls it, "before the excitation time" without substantive explanation; I'm not interested in Norton's "before the" part. I appreciate the fact that she pulled it out of his paper, but given the length of this video and wealth of preliminary crud in the first 4 minutes, I think a little more thought could have been given to a complete discussion and development of the concept, as the whole argument seems to balance precariously on its dome. Nobody in the comments seems to think the thing needs any _a priori_ development either. They just plug it into their own arguments like a well-understood given.
I'm struggling with the idea of T (excitation time)... if the original conditions was a ball placed on the apex of a dome with no external interference, then (in my feeble eyes) the implicit nature of such an excitation time seems to imply that indeed something does effect our ball. Also, the deterministic answer is r(t)=0, but for what interval? I am guessing it would be 0
If you look at the mechanics of deformable solids, you end up with lots of non-deterministic situations like this. For example a truss member loaded in compression will buckle when the load hits a critical value, but the direction in which the member will buckle is non-deterministic. To solve such problems, engineers often have do bias their numerical simulations slightly to force a given bucking direction so that the simulation can run without crashing.
Is this non-deterministic or rather you lack all the information to determine it? Like internal structure, steel crystalline lattice of this specific truss etc which makes it buckle this or another way.
As a structure engineer, we describe buckling as a harmonic of nth order, where n is the number (how much) the system’s statically indeterminate. (For linear elements with constant cross section) As far as i know, programs output all the possible buckling outcomes, however, you are right, in order to force a ubiquitous bucking direction, we must “lean” differentially little on one side.
5:00 Why should I accept that it's possible to nudge the ball exactly the right amount to get it to the top? I feel like it's weird that this is just getting glossed over, and I'm hoping this gets addressed later in the video, because my intuition is that you always arrive *at* the apex with a nonzero velocity (which you need to move from one place to another) and that the velocity you came in with will *always* carry you over, if you make it to the top. You can get infinitesimally close, but not quite actually to the exact point.
I think that is the critical flaw in this paradox. It's _not_ physically or mathematically possible for the ball to stop moving with zero acceleration, and at the exact top of the dome, there is zero acceleration (else it would be impossible to balance the ball there in the first place). Unless there is friction, it is physically and mathematically impossible for a rolling ball to stop rolling at a point where it could theoretically be balanced. Balancing the ball requires zero acceleration. Stopping it requires nonzero acceleration. The same point cannot fulfill both conditions.
If you made a mathematically perfect version of this dome and give the ball a push perfectly upwards with the perfect velocity, then it will indeed come to rest at the top indefinitely within a finite time. (Or at least that outcome would be completely consistent with Newtonian mechanics)
I think, for me, the idea of "infinitesimally close" is at the heart of all this. There is a basic assumption that there IS (or should be) a point where the dome is effectively horizontal and the ball can rest there. In fact I think that may be impossible to reach, not just in reality, but in the maths as well. You can never actually be in equilibrium here, you can only ever be infinitesimally close to it, hence there is no actual equilibrium. It's bothersome to the human mind but I think it may be true.
I'm not sure I buy the argument, but it did get me thinking about trying to push the ball up the hill at such an exact speed that it will stop there, assuming a completely smooth surface of the dome and the ball, i.e. atoms aren't a thing. If the speed was off by just 0.000000000000001 m/s (or less) the ball would appear to stop for a period of time, and then begin rolling again. The smaller the error, the longer the ball will appear motionless (but is actually moving infinitessimally). I know this isn't the thought experiment discussed, but it's still interesting nonetheless.
I was think along the same line. If we could put the ball arbitrarily close to the center, then the ball could seem motionless for arbitrary amount of time before rolling off. It is still theoretically possible to get the exact speed up hill so that the ball stops at the apex. and there are multiple paths and time this could happen. If we accept that Newtonian physics is time reversal, then it becomes nondeterministic.
I first thought it wouldn't stop at a finite time. Because at that moment i thought it was a normal spherical dome. But the shape is chosen in a way, that with the exact starting condition it will stop after a finite time. And every other path is limited by that time too! On this dome the ball can never appear to stand still. The only way it can stay up longer is if it rests exactly on the top for a while.
E.g. if you start at r=9, The ball reaches the top at t=6. And it is back at the starting point at t=12 if it starts rolling immediately. If you get the starting speed slightly wrong, the ball is back at the bottom after less than 12 time units. No matter how close you get to the exact value.
i think in ideal world where we can be perfectly align it, even if u take the side it doesn't role on its own u would also have to admit that such movement is no longer time-reversal. for me i feel time-reversal is much important core aspect of newton physics then determinism, being time-reversal lead u to think its deterministic which make sense, but here we have example where time-reversal isn't deterministic which disprove our claim that time-reversal mean deterministic.
@@FernTheRobotyou just made a great summary of the whole thing. I think 🤔 If it were possible to place the ball perfectly at the top, mathematics says there’s only one way to do it. Which is kinda surprising
If I'm at the North Pole, which direction do I go in to head south? If I'm even a millimetre off, there is only one south. If I'm precisely at the North Pole, every direction is south, so I sit there forever, frozen by indecision (and the cold weather...)
Sorry for your indecision . . . Really this is a case of if every answer is an acceptable answer. Down select if you prefer walking or boat, and start moving toward your goal. Still can not decide throw a pencil in the air and start moving in the direction the penciled lands pointing. Usually "good enough” is "good enough". The siting frozen is the 1 decision possible that will prevent accomplishing the south pole goal. And once you get to the south pole, make a snow angle, so you can be sure the point that is the exact south pole, will be withing a 30cm x 30cm box that is your chest.
@@jamieclarke321 its the background music used that resembles another's youtuber usual use of background music, that of Spirit of the Law, a mathematician who analyses the Age of Empires II game and the franchise.
There is a divergence at r=0. It means the description at this point is unphysical. Nothing to philosophize about it, in my opinion. It is somewhat similar to physics of shock waves. In the past, people thought there cannot be a sudden step jump in physical parameters at the front of a shock wave. Which is true. Therefore, many considered the shock wave itself as non-physical. Later, people learned that the change at the front is gradual on the molecular scale, which seems sudden on the macroscopic scale. So no problem there. Today the theory of shock waves seems well established.
They said that Newton is non deterministic. But they based it off of a velocity equation that was non-deterministic. So you can't introduce non-determism and then say well this must be Newton's fault and not my own.
Having a divergence isn't unphysical. General relativity has divergences all the time that disappear by coordinate transformation. In this case, you describe the acceleration in terms of the distance, but why not in terms of the time (in which case it disappears)?
@@duncanhw I think that generally in physics a divergence means that some other physics is missing that is smoothing the effect. For example, if I remember correctly in GR there are divergences in black holes that are smoothed by quantum effects.
Thanks for pointing this interesting problem! As a theoretical physicist, I wanted to better understand, so I did some analytical calculations that gave me another picture of this problem. First, to have a better grasp, it's good to consider all kinds of domes (2d cross-sections are sufficient) with different shapes, and more specifically around the top which is where the problem resides. For usual domes, where h(r) is at least a C2 function around r = 0, one finds the usual behaviour: a ball at rest at the top stays there, because of such initial conditions that determine the future of its behaviour. When this condition is broken -which is the case for Norton's dome but also for a whole family of domes, it can be shown that the same kind of trajectories as those found by Norton exist when the ball is at rest at the top. The indeterminism is not shocking me more than the situation of buckling, where the moment of buckling is not specified by the equations: this would mean that something is lacking in the problem formulation to determine T. What was not considered up to now is that such domes have an infinite curvature (or zero radius of curvature) at their top, which explains why a movement can follow rest at the top, specified by the usual initial conditions. This, for me, is the main surprise of this problem, where such situations are not usually considered. Experimentally, this is difficult to really observe this phenomenon as a finite ball, friction, perturbations, and precise initial conditions are all a problem compared to the theory. However, one interesting thing coud be to try, for equal conditions, several types of domes to see how the behaviour changes with the dome shape. Answer me if interested!
@ernesthector2451 I think the thought experiment makes no sense in theory at all. The only flat surface mathematically is a single point considered 0. Every ball that exists is bigger than 0, even if just an attometer. It, therefore, would be subject to curvature. Even if just millions of zeros behind the komma. It is just a matter of time until we can perceive movement, but it would happen at the very moment the ball is placed on the apex.
Genuine question here... I don't understand why the solution to the dome has two possibilities and one of them includes an excitation time. Shouldn't there be only one solution? And if you place the ball at the top of the dome it will just not move? Ever. I mean, if we were REALLY in the vacuum of space and no forces acting on it other than gravity towards the base of the dome. If there is an "excitation" at all, shouldn't the excitation be external? Again following all the laws? In real life if the ball stays still for a moment and then starts rolling by itself, my intuition would tell me that there was an external force. Wind? Small vibration imperceptible to us? Anything that we can't see cause it's too small or weak, but at microscopic levels, it happened. And it's weak but strong enough to remove the ball from its unstable equilibrium point. Gravity does the rest. So the bottom line is... I don't understand why there's a solution at all where the ball moves spontaneously. Can someone explain?
@sabapc81 SO... There are many problems in what you have said. You are hardheadedly applying an intuition that has been called into question in this very video. The intuition that "on a ball sitting on the top of a dome will not move". The real life argument is again a problem: real life follows a mix of relativity and quantum mechanics. Even better: we have no idea what real life follows and quantum mechanics and relativity are our best approximations. You are applying a model a little bit hardly on to a reality we do not know how behaves at plank's distances. The key point is 3:10 this is a model, and we are working with the model and testing its limits. The second key point is 4:41 where she states time-reversal symmetry and thanks to which if you accept that you can nudge a ball on top of a dome then you get that the reverse is a solution. There are already two solutions here: the solution of the ball that has been sitting on the top of the dome since the dawn of time and for the rest of eternity and the solution of you nudging impeccably the ball onto the peak of the dome. This is the precise key point because if the dome was perfectly SMOOTH then the resulting vector field for you differential equation would be smooth or at least Lipschitz continuous and then you actually cannot nudge a ball on top of it. No amount of energy would allow you perfectly put it on top: you either undershoot it or you overshoot it. Which now that I think about it may also be counter intuitive: you cannot nudge a ball to stay perfectly on top of a dome in smooth conditions. Consequence? Uniqueness of solutions. Excitation is just a funny name: there are no external forces. This is just math. I am trying to stress over it again and again because it is not clear to me which is the problem and what is exactly the point that should be addressed. You have said "In real life if the ball stays still for a moment and then starts rolling by itself, my intuition would tell me that there was an external force. Wind? Small vibration imperceptible to us? Anything that we can't see cause it's too small or weak, but at microscopic levels, it happened. And it's weak but strong enough to remove the ball from its unstable equilibrium point. Gravity does the rest." What I see here is "But in real life the models tells us this" and it is good when you test the model but at the moment you are sticking to the model like a religion. "It has moved, the theory that I have been taught tells me that a force was applied. Where is that force?" What your brain is grasping at is a mathematical model that we have glued on top of newtonian mechanics for years. Forces, energies, momenta, potentials are all part of the model, a "smooth" model. All concepts make sense when all the quantity involved are smooth. It the forces are only continuous then we have what this video is all about. Peano theorem grants us existence and not uniqueness for only continuous vector fields. The video is meant to break your intuition because your intuition was wrong, like ours. Welcome to the club. You firmly believe that the ball won't move. The way we have modelled the theory says it can. We have no experiment to make that can disprove that this case can or can't happen. Either our way of modelling newton mechanics is wrong and all the concepts defined there should be reworked to accommodate this example, or newtonian mechanics needs to have an assumption of smoothness to properly function. I hope I made something a little bit clearer and this was not just a random rambling of sorts. I apologise, it is 4:30 in the morning.
@lolmanthecat Thanks for the effort.in explaining. I will watch the video a couple more times. Not quite understood but I won't into it further. It's late here too.
Your intuition is that in the absence of an excitation, an object at rest will stay at rest. But this implies that an object in motion, under these conditions and subject to only constant forces, must always be in motion. This implication is plainly contradicted by models of straightforward situations using standard physics rules. According to the paper, you can use a purely deterministic set of rules to create a nondeterministic result, which contradicts your intuition.
Imagine going all the way back to the beginning of time, or what Aristoles called the first cause, an unmoved mover. How did this excitation happen without anything else? The problem lies within our intuition that the math here tries to counter. So yes as absurd and abstract as it might be, math is still math and it doesn't have to make intuitive sense, but it can bring us somewhat closer to an indetermistic answer.
I am an engineer, I am not so heavily math inclined anymore but two of my favourite courses, gave me memories that are somewhat aligned with what this video discusses. 1. Buckling where a beam was assumed to have a force perfectly acting along its axis, compressing it and we induced a small pertrubation to make it buckle, this pertrubation was explained to us in the fact that no force is ever perfectly vertical and even if it is, there is no beam that is perfectly symmetric and no material that is as perfect as we imagine while studying. 2. Oscillations, where the first thing the professor said is that we are never solving a physical system, but rather a model that tries to reflect the behavior of said physical system, if the results are bad, the model is bad for that specific physical system, but not necessarily for any other. This is why for simple problems we can ignore air resistance, but if you do that when studying the motion of a plane, or a fast moving car, your results are very, very wrong. In the dome's case, in the real world, the buckling arguments apply there always might be a small pertrubation, from the wind, or from the geometry of the dome itself not being perfect that will eventually cause the ball to roll down. In the idealized world of math, both solutions are valid, the stationary and the rolling one, but Newton's laws cannot be used to explain the transition phase because they are a mathematical model and the function that describes the ball's is simply an edge case that cannot be so elegantly explained due to the way those laws are built. I am looking at this argument completely cynically since it is my job to identify areas when a particular model is applicable, when not and why, admittedly taking Newtonian physics as the foundation of everything, but I can perfectly imagine an external factor, one we know exists, or existed, but are not worrying ourselves about how or why, being responsible for the definition of the excitation time T being an arbitrary value, just so the model can make mathematical sense, just like how a small pertrubation causes a beam to buckle, one that can be explained in the physical world, but we do not concern ourselves by defining further, just so we can get the solution we are looking for. So possibly the idealized world of Norton's dome is perfect compared to our world, but just imperfect enough to allow for this small nudge the ball needs, in a way we do not care to explain where it came from, so we just reduce it to a time we arbitrarily call T. No matter how ideal Norton defined this world to be, the existence of that time T is based on that imperfection, at least in my eyes.
@@comedyman4896wow, that's super helpful and insightful of you. Do you have any other Marvels of your intellect to drop on us? Is it commonly held piece of wisdom that if you don't have anything helpful to say, you should just keep your mouth shut. You aren't impressing anyone
@@comedyman4896it does answer the question to me. The dome equation says the ball "could start moving" but it's not proven and cannot be disproven. The answer is we don't really care because in real life it will eventually start moving and we know that, at which point Newtonian physics take over. I'm no expert but when looking at equations that are so large that the dome (paradox?) could add to a significant error if not accounted for, aren't other assumptions that are typically taken with a Newtonian approach likely to generate a far bigger error?
Assuming that the ball and dome are perfect, assume that the ball is perfectly placed, and it is in a vacuum and positioning such that there is no vibration to affect the ball. Please explain how the earth's rotation will not act upon the ball
@@sordidknifeparty Why don't you apply your own advice to yourself? You didn't even understand the point the other person was making, you certainly aren't impressing anyone.
usually if something breaks physics it's because it starts with flawed assumptions or flawed equations, at 18:33 is the possible choking point for flawed assumptions/equations, maybe it needs to be approached from a different angle
I know right, isn't when t < T then acceleration is negative so the ball is infinitely slowly slowing down right up until it stops and then infinitely slowly speeds up. When it is at actual zero depends on how closely you look at time. Like what is the acceleration of the ball that hits the wall at the exact moment it hits the wall? infinite for an infinitely small time. It is just a break down of Newton at small times and scales and we know it breaks down on that scale already because of Quantum. I think the "special" solution is just kicking a ball up a hill so it stops at the top and then falls back down. With an infinitely perfect ball and hill there is no point in time when the ball stops unless time tends to infinity small. So the ball is slowing down or it has always been there not that deep.
Some possibilities: What are the elasticity properties of the materials involved? Is the ball absolutely stationary at the start? Not just "as far as I can see!" Is the experimental set-up totally insulated from external vibrations, gravitational attractions from the operator, Etc...etc
I was looking at 17:00 the problem is reduced from motion over time to acceleration during an instant. IMO, if you speak about instants you discard time. It's like a function can be discontinuous in a point and still call it continuous because the limits on each side are the same.
I am at 1:30 in the video, my guess is that the complaint has to do with non-uniqueness of differential equations, here there's a failure of Picard Lindelof theorem induced by the singular point of the dome.
It's that plus the fact that after he concludes a non-falsifiable statement from this, we have hard proof and tons of papers that "no one has ever disproven it" 🤣
One thing I love about this is it puts “quantum weirdness” (and the competing interpretations of quantum theory) into perspective: it’s not that Newtonian mechanics is “intuitive” and quantum mechanics is “unintuitive”: it’s that Newtonian mechanics is intuitive *enough of the time* to put us off the scent of the ways in which it is unintuitive
No. The classical case is fundamentally different, because you need unrealistic fractional power force laws to get these types of non-deterministic behavior. The person who discovered this is James Clerk Maxwell, in the 1870s, his example is a particle which feels a force which goes like x^(1/3). This is exactly the same as the "Norton hill", so Norton is committing plagiarism, as usual for philosophy.
@@annacoeptis It's definitely the case. There is no example in nature of a non-Lipschitz force-law of this type, nor can you construct it on a microscopic level. Further, in quantum mechanics, there is no more ambiguity in these types of force laws than in regular force laws, they are both the same. This example is out of date and also not due to Norton, so it's plagiarism, and not even clever plagiarism at that.
the first quarter of this video asks all the right questions in order to be a very “self conscious narrative”, i mean, it feels like a near perfect beginning of a topic. very well thought and executed, kudos.
Newton did not take these types of edge cases into account. The theory is completed by calling that all derivatives of the velocity be equal to zero if the external forces are zero. In this case, the derivative of the acceleration or second derivative of the velocity is not zero for t>T. Therefore the ball does not stay still, Newton's law is not broken, but this implies that there is a disturbance at some instant T, which makes the second derivative of the velocity different from zero, leaving the rest case. Therefore the ball does not move spontaneously. Sorry for my bad english.
Newton stated the 1st law, because it is independent from the 2nd. That's why there are three of them and not two. The example in the video basically shows that Newton's theory is incomplete without 1st law, since the equations of motion from the 2nd law give multiple solutions. It's just so happens that in many problems only the 2nd law is enough.
i believe that is the reasoning behind "we need a 4th law". if the net external forces are zero, all derivatives of velocity must be zero. but once we do this, then we can ask what violations of this 4th law would look like and create experiments that would indicate 4th law violations.
what you're saying is... the 3rd derivative (called the *jerk* ...lol) is not zero... so eventually the acceleration changes from a zero to a non-zero value... that makes a lot of sense to me... but i think that has something to do with the reasoning behind needing a fourth law
this is also exactly how I think about it. I posted similar reaction separately yesterday, just saw your remark. only difference is that I started with the second derivative of the accelaration as that goes from 0 to 1/6 at the moment the movement starts. so that should be the rootcause.
Great video, but can someone help me out? At 18:33, she equates t and T, which means the power is 0, and then she says "the whole *thing* is 0" - surely it's 1? What am I missing?
@steviestickman680 Right, thanks. If there was a derivation of the formula, I might have spotted that - on the other hand, it might have gone over my head and I'd have not finished the vid.
That made much more sense than it should... I mean, I do not subscribe to determinism, neither physical nor philosophical, but it is counter intuitive to think about Newtown's laws in an indeterministic way! Very cool!
one can argue this solution is not physical using an argument from calculus. while location, velocity and acceleration are all continuous functions, the derivitive of acceleration, also known as Jerk, is 0 when t
I don't agree that for physicality the acceleration or even less the jerk needs to be continuous. Consider a block sliding and coming to rest under simple dynamic friction. That example has a discontinuous acceleration, and a Dirac delta -like jerk. I think the conclusion from the video is that Newtonian mechanics simply is not always deterministic.
Do we actually have reason to believe jerk is smooth always? This is something I have wondered a long time. Certainly solutions are not analytic since at some point objects sometimes stop.
But doesn't that argument fall apart for the time reversed situation out the ball rolling up the dome and coming to rest at the top. No-one has any argument with that being a valid solution of Newtonian equations of motion, but it also has a discontinuity in jerk and divergence to infinity at the point in time at which the ball comes to rest.
@@ahhuhtal Well, we *do* know that "simple dynamic friction" is an approximation for a perfectly-rigid body - which, of course, is untrue. At lower jerks, we know 'softer' things (e.g. tires) have additional behaviour beyond that, which smooth out these transitions by deformation. It should be the case that at larger jerks, even the most rigid bodies in reality will deform. Take the idea that rigid bodies transfer forces instantaneously. In reality, they transfer at approximately the compressional velocity of sound in the material. It's just that for a sufficiently rigid body dealing with forces on larger time-scales, this is a fine approximation, but if you start pushing at the edges of this theory, it doesn't truly work.
I've never made a TH-cam video for the general public, but I can see that an in-depth explanatory video like Jade Tan-Holmes (of Up and Atom) makes requires an enormous amount of work and research. Yet she is always so fresh.
She did NOT do her research! If she had done her research, she would have discovered this is a 19th century argument by James Clerk Maxwell to explain the "Lucretius Swerve" of ancient poetry. Norton is plagiarizing the argument, gambling on the public being too ill-read to find the original source.
Not-a-physicist here: From my understanding, I don't see where that breaks "symmetry". Newton laws still apply if running things in reverse, at the very least with a proper value for big T. If you were to perfectly push a perfect ball on a perfect dome so that it stops perfectly on the apex, it would stay there for an infinite time. In reverse (or forward, if pushing the ball was the reverse) the postulated position equations (for t = T) are still valid as long as T is infinity as well.
I like how all this intuition is time-invariant, that any insight here will also apply in reverse. It makes much more sense to understand the excitation moment as the ball coming to a rest at the top of the dome when its kinetic energy becomes zero, and how our understanding of acceleration at that time approaches nonsensical. It makes sense that there would be no "last instant" of the ball accelerating to the top of the dome - this is just another examples of the Achilles paradox.
Another great piece of work. Thank you. The edge conditions that break a model are very useful for at least 2 reasons. First, they show where the model can be improved by understanding the underlying reality better. Improvement on models leads to new physics which leads to new models which leads to new physics... Second, at these boundary conditions where nature doesn't normally go, we find very useful and non-intuitive uses of physics to invent things we never thought possible before.
Particle physicist here: lovely! Very nice. I love dropping by your station from time to time to see what's new. Always very curious,novel stuff, competently and lovingly presented. best regards, DKB
Not quite sure if it's the same problem as the one I'm thinking of, but a similar thing can happen in stress buckling of a cylinder under pressure. For an ideal cylinder (e.g. a soda can) it is impossible to predict where it will buckle, since the initial conditions needed for a prediction are the same all along the radius. It's an interesting quirk of simulations that they typically terminate before buckling occurs, since the solution is indeterminate and breaks the linearity assumptions baked into most FEM programs.
it is perfectly possible to guess where it will buckle in the real world though from inspection of the metallic grain structure. The real world does not have truly ideal cylinders that are perfectly strong to resist compression forces in all directions perfectly, so it's an irrelevant and useless thought experiment. The universe isn't made up of mathematically perfect surfaces.
That is not analogous to this, its comparable to how Newtonian mechanics will say that a ball at the top of a hill wont roll down, but irl it will roll in some seemingly arbitrary direction, but this is actually due to it not actually being exactly on infinitesimal top of the hill and a variety of other factors like temperature, wind, and things having non smooth atomic structure. From a theoretical pov, the ideal cylinder should never buckle, just as from a theoretical pov, an ideal stationary ball on the top of a hill should never roll. The difference here being that there is a challenge to that theoretical conclusion.
8:25 Your impersonation of me watching the beginning of most your videos before you break things down it's spot on! I've probably said it before but its worth saying again, your ability to break these highly complex topics down for us in a way that's as in depth as it is entertaining is truly a gift and I'm grateful that you choose to share that gift with us!
If you are ignoring friction, then any speed that can reach the apex will no longer have any resistance to its motion as it moves to the horizontal instance of the apex and so will continue to the opposing slope. If it had less energy it would not reach the apex, there is no perfect speed as to transition from the slope to the apex is to have some remaining speed which is no longer resisted, so it is impossible to land perfectly at the top of the dome though the process described. to be otherwise is to introduce some mechanism that would in reverse explain the behaviour.
Yeah, the real paradox here is the idea that a rolling ball can stop rolling with zero acceleration. That violates Newton's first law. The acceleration due to gravity goes to zero as the slope of the dome goes to zero. It is mathematically and physically impossible for a ball to stop rolling at the precise point where it has zero acceleration.
@@FirstLast-gw5mg Since there's no friction calling it a 'rolling' ball introduces an unneeded and obfuscating parameter. It's just a 'moving' ball. And it doesn't even need to be a ball, it could be a mummified toad.
@@paulkline515 You're mostly not wrong, but I'd clarify that it can be any object that has exactly 1 point of contact with the surface. If the object has multiple points of contact with the surface, I think it could possibly end up saddling over the top.
You roll the ball up the dome such that the initial kinetic energy is equal to the potential energy the ball would have at the top of the dome. Newton's laws don't forbid this.
@@FirstLast-gw5mgit is certainly possible, theoretically speaking, to roll a ball up a frictionless hill and have it stop perfectly at the apex with gravity being the opposing force. Looking at just the mathematical model, it just means there is only one solution at any given approach angle that allows for this.
We have to ask ourselves what the first law of Newton actually says. At 17:19 Norton reformulates the first law so that his proposed solution of the e.o.m. doesn't violate it. The problem is that his rephrasing distorts the actual meaning of the first law. Basically, it says a(t)=0 for F(t)=0. But isn't this just a special case of Newton's second law, which says F=ma ? Then the first law would be just a special case of the second law and therefore completely redundant? Was the great Isaac Newton really that sloppy? Short answer: No. The first law must say something different. Let's look at it again: "Every object perseveres in its state of rest, or of uniform motion in a right line, except insofar as it is compelled to change that state by forces impressed thereon." (source: en.m.wikipedia.org/wiki/Newton%27s_laws_of_motion) And now apply it to the dome: Assume a ball to be perfectly on top of the dome. Unless it is slightly dislocated by a perturbation - which we don't assume to be the case - the net force on the ball is zero. Therefore it stays at rest. If the ball is dislocated from the equilibrium point (i.e. the top of the dome) then of course there is gravity exerting a force on the ball which causes it to move. But what Norton doesn't get here is that Newton's first law prevents such a dislocation from the equilibrium point. Therefore Norton's proposed solution violates Newton's first law.
You act like the laws of motion aren’t redundant, but really they all are just simplifications of special cases of the second law. If force is zero an object will not accelerate. That is the first law in terms of the second. There’s another example though. The third law states that forces are equal and opposite. This is just the first law if you think about it though, because if you take an isolated closed system with only two objects that are exerting forces on each other, the forces must be equal and opposite, otherwise their center of mass would accelerate, violating the first law, which is also violating the second law, since there is no external force on the system, and therefore no acceleration of the system. The first and second laws can both be completely derived from the second law.
The dislocation starts as an infinitely small dislocation. Since you can flick the ball up the dome and it will come to rest at the summit, you can always time reverse the equations. Now you will have another time Tf, the flick time. A human decides the flick time. From this you derive T. The real problem here is math and 0/0 analog.
This is just bogus, Newton's first law is known to be a special case of Newton's second law. All mathematical expressions of it can be derived from second law. And "he couldn't be sloppy" is just a bad argument in general.
"We are tempted to think of the instant t=T as the first instant at which the mass moves. But that is not so. It is the _last_ instant at which the mass does _not_ move." Fascinating stuff!
Pleasantly surprised to see Picard-Lindelöf on TH-cam. A lot of people are getting hung up on the real world/physicality and that Newton's laws are not suitably at all scales. That's not the point. The point is whether even the *theoretical* Newtonian model purely as a mathematical model is deterministic as most scientists take it to be. One way to understand this is to imagine that you want to program a perfect virtual world in a special computer (which can do perfect calculus, say) with a perfect Newtonian game engine and then test if this game engine is deterministic. To truly answer this, one would need to give a precise and complete formulation of the Newtonian model first: what "manifold" we take for spacetime (R^{3,1} might be good)?, what regularity of functions are allowed for creating the virtual world?, are corners and kinks allowed for the solids in the word (the apex of Norton's dome is not differentiable)? are there elastic solids, fluids? etc. The math in the paper is well known---it is not too hard to find scenarios where unique solutions break down. This means in our perfect virtual world admitting Norton's dome, the computer running the game engine will try to solve the ODE as it is instructed to according to Newton's 2nd law and then run into a conundrum because there are multiple solutions and it must choose one---deterministic or not, a 4th law must be added (otherwise the computer will freeze/hang here). We could program it to choose one randomly---even that is actually tricky---if there are infinitely many solutions we need to assign a reasonable "measure" on the space of these solutions. For a deterministic virtual world, the 4th law needs to help pick out a unique solution---probably tricker but I haven't thought about it. The simplest way to go is to just impose sufficiently high regularity (analytic, say) on everything (disqualifying Norton's dome) so that the ODEs never have uniqueness issues.
I’m kinda confused because the answer seems obvious to me and I doubt I have the answer if indeed the issue is as big as you claim. I will read the paper when I have time just to be sure: the equation written is using a mathematical “sleight of hand” as you put it. It introduces a discontinuity in the 4th time derivative of position (the snap). This is physically impossible, because you would need an infinite variation of the jerk so, just like in other cases when using infinities, it breaks down. Now, this is probably wrong but if anyone reads this and can tell me why I would appreciate it.
You're not the only person to comment this so I think you might be right. I wonder if any of those papers in the map she showed have this argument laid out in them.
In another comment thread, a poster pointed out that Norton provided the ball bouncing off a wall as one well known example of accepted Newtonian physics where the higher order derivatives are discontinuous. And have one more example - a ball rolling off a table and suddenly experiencing gravity
@@mengmao5033 I thought about the same example, however the discontinuous derivatives come from an idealized interaction with another body moving at different speed: we are not investigating the cause of motion because we are idealizing force transfer to be instantaneous. I also thought about the table, because at face value it seems different, however, when the ball is rolling off, you are ideally removing a force acting on the ball, giving a discontinuous derivative. We can accept these discontinuous derivatives due to the fact that we understand there is a force transfer and we are simply idealizing the “start up”, but it doesn’t work without interaction between bodies at different speed. At least, that’s how I thought of it
Really interesting video! Small note: at 15:19 the example with the ball colliding with the barrier, it's true that you get u^2 = v^2. However, it's important to note that u and v denote SPEED, the magnitude of the velocity vector, NOT the velocity vector itself. While it's perfectly fine to have one VECTOR be the negative of another, magnitudes can only be positive (or zero), so there's no such thing as having a negative speed. The solution u = -v is in fact the erroneous one, NOT by any physical intuition but instead because of how u and v are defined. The first solution, u = v is actually fine, keeping in mind that it only says that the initial speed is the same as the final speed-- as it turns out, this is the only thing that you can deduce from applying conservation of energy here anyway. The direction of of the ball's velocity afterwards can be found only by incorporating other laws, constraints, or assumptions.
The ball bouncing off the wall doesn't follow Newtonian physics anyway as that would involve the wall pushing back with twice the force of the ball hitting it and not gaining any momentum from the impact. This would be creating energy from nothing.
@@terenceundbud the magnitude of a vector is never negative... it is the length of a vector. At least in Euclidian space if you do not use imaginary or complex coordinates.
@@shinzon0 In a coordinate system, negative amplitudes(in scalar components) are used for vectors to choose the opposite direction in comparison of the unit vectors in the vectorial base.
You roughly say "some philosophers go as far as saying there is no one true version of Newtonian physics" at around 20:20. But isn't that particular point obvious actually? "All models are wrong, some are useful" seems to me to explain most of this conversation. I personally think the discrimination between these different models is inherently arbitrary because it's beyond humanitys epistemic reach.
Great video! Without speaking from any kind of physics background, I still think it makes sense. We create a scenario in which there is no force acting on the ball, but on the other hand nothing holding it either, not even friction, which means we have to remember that this is not a realistic scenario. Therefore intuitively it might as well roll rather than sit still. I personally find it really fascinating when someone can break our habitual thinking in such a simple way, whether it turns out to have practical applications or not🙂
Over 4000 comments in 15 hours (far more comments than seemingly most or all other videos on this channel) - this nicely reflects the mind-bending and emotion-stirring nature of Norton's Dome that I've loved about it for a long time. Congratulations on being [as far as I'm aware] the first major TH-cam channel to cover this topic!
It's not "Norton's dome", it's Maxwell's example of a Lucretian swerve. Norton plagiarized it from 1870s work by a physicist, as is typical for philosophers. The reason this is getting a lot of comments is that it is fascist-adjacent, Maxwell used this type of system to argue against full determinism in the physical laws. All these debates are obsolete since quantum mechanics.
@@bjornfeuerbacher5514 If Norton is supposedly a "plagiarist" (as Annaclarafenyo8185 has claimed in multiple places in comments here), then Maxwell was also a "plagiarist" by this definition. The first historical source for such an example is likely a paper by Poisson in 1806 (where it was mostly just conceived regarding the function of the force), though it didn't get a lot of attention until Boussinesq discussed it in 1879, which led to a flurry of discussion in French physics literature... and then migrated to discussion that was mostly much less formal and technical among other non-French physicists like Maxwell, who were theorizing about free will. Norton's example is really a special case of Boussinesq's "dome." This example was so well-known in the 19th century literature (especially French physics literature) that I'd go so far as to say it was "common knowledge" (which doesn't require citation). And indeed in Norton's initial paper from 2003, he doesn't present this as anything special or novel -- merely one of many shapes that will behave in this way. It's not really anything particularly exotic physically, and there were periodic "rediscoveries" of such shapes and functions in physics literature over the years (at least two different papers in the 1990s alone that I know of). For a summary of some of this history, you might look up the article "The Norton dome and the nineteenth century foundations of determinism" by Marij van Strien (2014). The emphasis there is on the French sources, since they discuss such things in more depth (to my knowledge) than people like Maxwell, whom I believe just used this as a vague example in some speculation on free will and determinism. As for who should get ultimate "credit" or whether "Norton's Dome" is misnamed, see Stigler's Law of Eponymy (which was actually formulated by Robert Merton).
5:00 I might have misnderstood the setup but this is not possible in perfect newtonian dynamics: the ball will either asymptotically go to the dome or surpass it. The differential equation that governs the physics satisfies the hypotesis of the existence and uniqueness theorem.
Thank you. My comment was "is it theoretically possible to roll the ball to rest at the apex of the dome?" I suspected it couldn't be possible, which makes this problem akin to slight of hand. If we assume the ball CAN come to rest on the apex, then we have to assume spontaneous, uncaused motion. If it CAN'T, there is no problem.
@@gochaosgamer1759I mean if it reaches the perfect spot, it would need slight momentum to get there which would just roll over. You can approach the top but never reach. Isn’t this just limits then?
@@Bigleypit is just limits. This video is such pop sci “chocolate and wine are healthy for you!1!” type stuff. It takes a bad paper and makes it consumable for the masses to think something magical. It’s a shame she’s always been such a sham of a channel like this when she could do so much better.
@@ClementinesmWTF It is not just a limit. The ball reaches a state of rest at the apex of the dome after finite time. That's how the equations in this case play out and you can verify this if you just do some calculus yourself. The differential equation does not guarantee a unique solution because it has been constructed to avoid one.
@ so the equation has an undefined/infinite spike at the very end? Ok? Cool? All you’ve shown is that you can make a dome with an equation that is non-analytic. Like…cool. That doesn’t have implications on newtonian physics other than the equation yoy made up is not permitted-which is fine as it wasn’t something you could do in the real world or even a Newtonian world anyways.
I had asked similar question to my science teacher when I was 11. What happens at the exact time when ball hits the bat and changes its direction instantly. I was told that it does “instantly” so we don’t need to take care about it. Later, I realised that science creates an abstract model which tries to get isomorphic to real world scenario. But there is no infinitely precise abstract model that can define reality. I sometimes think of a world where even the first order formal logic fails. What would it look like?
Well for that case, and basically all real cases, the answer is simply that it doesn't change instantly. There are technically electrically forces at all distances between the ball and the bat, and when it "impacts" they reach a peak and then subside, and the forces were completely continuous and smooth the whole time.
2:47 in and this seems like an in-issue; newtons laws describe perfect theoretical environments, which ours is not; the top of the cone is not flat, we are not in a vacuum, and the ball is never perfectly at rest, it’s moving microscopically as the minute forces around it, like air, accrue in an eventually-compounding direction enough for gravity to get its hooks in. Think of Brownian motion. Am I missing something?
What you are missing is that the argument is that the ball will spontaneously move off the apex at some time _in this theoretical world._ There's no point talking about realworld imperfect conditions when the topic of interest is this perfect (i.e. impossible) world of a frictionless ball and dome in a vacuum only being acted on by gravity. The point of the paper was to advance the idea that Newtonian mechanics can _also_ be indeterminate, that a Newtonian world is not fixed on rails. Personally though I find the argument unconvincing because crux of the problem relies on a quirk of mathematics that the equation in question has multiple possible solutions. It doesn't provide a mechanism for _why_ that excitation period happens when it does, it just supposes that it does. My instinctive go-to response is that the problem is ill-posed, not that there's a hidden truth in Newtonian mechanics (which as we all know, has been rendered obsolete, but is a useful simple model for general, non-extreme cases).
I'm not entirely convinced that it is actually possible to roll the ball up the hill with just enough energy to stop at the apex. I know it feels intuitively true, but is it actually? It an idealised setup with no friction, one could argue that the ball must always arrive at the apex with some non zero velocity and therefore overshoot it. I think it's a fair question to ask at least.
The video raises a fascinating question about reversibility. The ball can reach the top of the dome from many different directions around the base. But the crucial question is: if the ball starts at the top, can it roll down in the exact opposite direction it came from? If we placed the ball on top of the dome, that direction is "up," which isn't reversible. This relates to the idea of one-to-many versus one-to-one relationships. Many starting points (directions around the base of the dome) lead to a single endpoint (the top). However, when we reverse this, that single point at the top can lead to many possible directions. This "one-to-many" aspect makes perfect reversal impossible. It's like having one input (the top of the dome) leading to many possible outputs (directions of descent). Imagine a circle around the base of the dome. Any point on that circle represents a possible starting direction. All these points converge to a single point at the top. But from that single point, the ball could roll in any direction along that same circle. This inherent asymmetry prevents a perfect reversal of the trajectory.
well all those points on a circle, where the ball starts to ascend, converge on 1 point. So there is a one-to-many correspondence as well. This is the same thing as both 2^2 and -2^2 equaling 4, which results in sqrt(4)=2;-2
It's not all that fascinating. The math in physics is there to describe what we can observe happening. We cannot observe time reversing itself, so anyone who includes that as part of their discussion is playing silly buggers right from the start.
This is an interesting approach, but for me, this is not really how time reversal symmetry works. By this logic, kicking the ball is also not a deterministic action if we observe the situation from the ending point, because the ball lying on the ground can get there from many directions, even if we describe it with a seemingly identical equation. Basically, the foot kicking the ball can be placed anywhere on a circle, facing the center of the circle where the ball will be in a "resting" position. If we reverse the action, the ball shouldn't jump back to the exact same point, it could fly to any random point on this circle. The trick in this situation is that it is seemingly the reverse action that is establishing the situation. We claim that if we can push the ball up to the apex point, into a resting position with a perfectly executed impact, then the reversal should be a scenario when the ball seemingly starts to move from a stable resting state into a downwards rolling motion. The reversal is only relevant to explain why we observe the situation described in the thought experiment, the direction of motion is irrelevant.
@@filipsperl I used a similar approach, considering a Mobius strip. We can tell if a strip is a Mobius strip by how many sides it has, which is determined by how many half-twists are applied. We end up with a function like s = f(t), where s is sides 1 or 2 and t is any integer. If t is odd, s = 1; otherwise, s = 2. This function cannot be reversed because it is a one-to-many relationship; it cannot be determined by setting s to any value. Your approach is more concise and easier to understand. I wanted to use an example, but my Mobius strip analogy was too complex. I like yours better.
@@BotloB I get your point. Newton's cradle is a good example of a potential one-to-one reversal symmetry, which could support this argument if we remove macro-level variables from the system (as per the thought experiment).
Really enjoyed this! I think his argument is sound, and really quite clever. Another point I think worth emphasizing... his dome example actually DOES comport with our intuition. A ball on top of a dome just "feels" unstable. If you went to the trouble of constructing such a dome and such a ball to ridiculously exact specifications, placed the ball in the exact right position with exquisite care in a vacuum chamber in the dark, I think it's reasonable to expect it might stay there for little while... but we all kind of "know" that it would, in some relatively short amount of time, roll down. The conventional explanation is that, despite your best efforts, there will inevitably be some perturbation that upsets the balance just enough, e.g., an uneven momentary change in temperature that causes an uneven expansion in the body of the ball, or a passing photon, or any number of other physical things that could change things "just enough". And, sure, that's a fine explanation, I have no problems with it. But such things are close to impossible to measure, and thus close to impossible to prove. Seems like his solution is every bit as valid an explanation, honestly.
This also breaks the Curie principle since the rotational symmetry of the dome should't allow an explanation as to why the ball rolled in any particular direction instead of any other. This is profundly problematic as Newton's laws are invariant under rotation (otherwise you would break angular momentum conservation). Big trouble in here
Yep, when you assume than in ideal mental scenarios things start to move by themselves, specifically contradicting the rule set of that scenario stating that should not happen, all falls apart what a surprise pfft
@@freshrockpapa-e7799 The behavior of the ball directly contradicts Newton's first law...which is a big problem in a newtonian scenario And the mental gymnastics the author engages in to try to avoid the contradiction are quite lame Now repeat with me: Po-ta-to
@@TheChzoronzon Are you not only going to stalk my profile, but follow me and reply to me in every comment I make? Please get a life. Btw you don't have to waste your genious brain in the TH-cam comments. I'm sure you know about this open problem you just learnt about (and haven't even read the actual paper) more than the entire scientific community has figured out during the last 16 years, so go publish your own instead of wasting your time here.
There is already proof that 0.999... (infinitely many 9's) definitely equals 1. There is a Wikipedia article about it - including the proof: en.wikipedia.org/wiki/0.999... but with any finite number of nines - it's not 1.
The starting point is not zero acceleration/zero gravity. The answer is that these states do not exist in the physical world. Therefore maybe zero should not exist in the equations? It always confounds me that 5x0=0 Where did the physical go, it always exists, maybe in an infinitely smaller amount.
As the ball rolls up the dome, there is information about its position and momentum. After it comes to rest at the top, we know its position, but nothing about its history. Does that count as "information loss" (which we've been taught is impossible, leading to questions about information being lost in black holes)?
Just as there is not first point in time when it starts rolling there is also no last point in time where it stops rolling, the speed just is lower and lower the closer to the apex the ball is, it will never reach zero. It will never come to rest, the shape of the dome is chosen with just this attribute.
This is super cool! It’s of particular note that the Picard-Lindelöf theorem and the Lipschitz condition were formulated more than a hundred years after Newton, when mathematical analysis was really picking up steam as people were trying to resolve other mathematical paradoxes that seemed to show up, including many debates over whether Fourier was out of it for interchanging limits and integration. The result of these debates was mathematicians developing a lot more rigor over proof methods, and arriving at much more clarity of what was underlying points of contention-for example, for Fourier’s supposed indiscretions, it turned out to be a matter of which norm on function space one was dealing with for the limits, which turns out to stem from some really deep math on the topology of metric spaces-an entire field which didn’t exist as such in Newton’s time! So, an interpretation is: Newton’s linguistic description of his laws may have to be fiddled with-but that fiddling depends on mathematical nuances that weren’t worked out at the time! Operationalizing mechanics means choosing an interpretation in a particular mathematical structure (ie, with or without the Lipschitz condition required), and while both are valid, we now know from analysis that the resulting qualitative behavior of the physics-and hence how one would linguistically describe the physical laws-depends on that choice.
I think the problem is assuming that the ball would rest on the dome at all. If we can subdivide time to the point where there is no instant where the ball BEGINS to move, just the last instant where it is not moving, I think it follows that we can subdivide space as finely so that there is no way in which X amount of energy can be applied to the ball causing it to roll up the dome and stop at the top. Since it cannot stop at the top, it cannot roll back down. Certainly a ball can roll down a dome, just not from the apex. The ball (which only contacts the dome at one point) and the apex of the dome (itself only a single point) can never actually be resting on each other because they are both infinitely small. Since they can never meet due to our ability to subdivide ideal space so finely, there is essentially NO apex, and if the ball does seem to be at rest it merely a case of not being able to measure the motion because it is infinitely small. This practically required by saying "there is no moment when the ball BEGINS to move". There is no moment when it begins to move because it was already moving since it is impossible to balance an ideal ball on an ideal dome if we allow for infiite subdivision of space.
Same conclusion that I came to. 1) some one would need to do the actual maths, but with a smooth frictionless dome, any momentum remaining as you reached the "apex" would be retained and you would roll off the far side. 2) even if it appears "perfectly" centred there is a infinitesimal level of gap between the ball and the apex so it may even remain there for years (due to the miniscule level of acceleration that close to the top), eventually it will roll. (and with a dome shape the acceleration will transition from imperceptible to "sudden" as though it started magically)
@@bobcostas9716 I just ran it through chatgpt and it agrees with you and says that the time equation diverges when theta (angle difference of the object and the vertical line going through the center of the dome) is 0 meaning it takes forever until the ball fully rests. Its AI so it could be wrong
Surely it depends on whether there is a shortest unit of time (and everything jumps to the next place instantly). If there is then this is wrong I think and the start can be measured. Or have I misunderstood? Great video by the way
@ who knows? At some point there is either an analogue or digital underlying time thing going on, if that analogy works? Depending on which determines the dome paradox doesn’t it?
We learned about something like this in my mechanics 101 class several years before this paper. The example we had was a bucket with a hole in it. Eventually, the water drains into another bucket. But once it's done, you can no longer figure out WHEN it finished and a time reversal is no longer possible.
Yeah this relates deeply to the Arrow of Time and such... the whole premise that Newtonian processes are reversible has no basis. Not to mention that we already have so many other proofs for Newtonian Physics being incorrect about numerous things.
@seedmole The equations of Newtonian Mechanics are time reversible if you are allowed perfect precision in time and space, but not allowed infinitesimal division of time and space. But when you get anywhere close to the limits where that matters, you run into quantum mechanical limitations anyway.
but you can measure the hole, you can measure the volume of water, you can figure out how long it would take for the bucket to empty. time is a human concept. when it started, when it ended. even if you could say when those things were they are relative concepts based on when we have defined them based on the our position around the sun. For laws of physics the exact time the bucket was empty is meaningless. its not important at all and you can calculate every other property of the hypothetic scenario. needing to know exactly what time it was when the bucket emptied would be like saying you can't know how the water felt when it was emptied out. plus its a hypothetical. it emptied at whatever time the person made up that it would be empty.
As an observer of the macroscopic state, you wouldn't be able to infer when the water had finished draining, but if you knew the microscopic state -- the position and momentum of all the molecules involved -- then you could in fact make that inference. Run time backwards and the water in the lower bucket would sit there for some amount of time and then start leaping up and passing through the hole into the upper bucket, all without violating the rules of Newtonian mechanics.
If you rolled the ball perfectly up hill, i don't see how it can stop at the apex. The instant before the ball reaches the apex, it has velocity. At the apex, there are no forces on the ball to stop it there, so it must continue past the apex? What force acts on the ball to stop it exactly at the apex?
Very thought-provoking. The idea of time-reversing the ball, stopping it at the top, was what caught me. Infinitely many paths lead to that position, but it could also be the 2D cupula, where the top can be reached from each side. When the ball reaches zero speed at the top, the history of how it arrived there is lost, and all the driving forces (gravity and inertia) are also "erased" (balanced or zeroed). It reminds me of the information conservation principle in quantum mechanics, only in this ideal case, the information is really lost.
The information is not lost. To get a valid, complete Newtonian system, the dome is resting on a surface (as described). If you give a ball at the bottom some impulse so it comes to rest at the top, then you also imparted some force on yourself in the opposite direction (not to mention you yourself are also at some place next to the dome, revealing the direction it came from). And you then probably also imparted an impulse to that same surface the dome is on (or not and you float in space away from the dome) (presumably a planet generating the gravity?). So the whole planet ends up rotated after the ball (and you) have come to a rest (not sure about the resulting angular momentum of the planet though). No, I think the real issue is when, the STARTING condition is that the ball is already resting on the dome and then it suddenly rolls down to some direction.
An alternative is when the STARTING condition is when 'the force applier' doesn't exist but the ball starts at the bottom of the dome but with enough velocity already to reach the top. In order to slow down, it necessarily pushes on the dome, which then pushes on the surface (planet?), which starts rotating in some direction (hence, the direction is 'remembered'). I'm not sure if that rotating surface means the ball can't sit still on the top anymore for an arbitrary length of time. Could the surface end up with just a translational motion instead but no angular momentum?
@@peterbonnema8913 okay, set up four domes with each perfectly scented at the corners of a square. Then place pairs of balls in the exact centres of the left and right sides of this square where each ball in a pair can somehow impart the necessary force upon one another to have them move apart along the side of the square and up the domes such that they are centred perfectly on their respective domes. Then you have four balls on four domes with no way to determine which direction each of the balls came from to get onto each dome. This removes any way to definitively track the fource that pushed each ball onto the domes and it also removes any rotational forces imparted on the planet from the initial push. I'm sure the external world can be set up in a way to equalise gravitational forces on the balls, one way could be to have the whole set up be a cube in empty space with eight balls and eight domes instead of just four, that way the gravitational forces on each ball are all pulling equally towards the centre.
Mathematically speaking the only way for the ball to stop at the exact top is if it is still under some form of acceleration. The acceleration due to gravity goes to zero at the exact top, so the only way it could stop is by friction. You cannot roll a ball up a frictionless dome and have it stop and balance at the top. It is mathematically and physically impossible, and produces the exact contradiction in physics that this theorem represents.
Math does not rule physical phenomena, it merely describes our observations and guides our predictions. The finer our discrimination of differences becomes (the better our detection systems are), the tighter we must control initial conditions for the math (I,e, our model) to match the observations. I am troubled by time being a parameter in explaining the dome problem. As I see it, balancing a ball on top of a dome requires a point solution. Even if it were possible to achieve that precise point solution balance, the real world has perturbations outside of our ability to discriminate. Any unmeasurably small perturbation (Geologic, atmospheric, etc,) will upset the point solution and the ball will roll off the dome. My experience leads me to believe that said unmeasurable perturbations WILL occur. Predicting when such perturbations happen is the only role for any time parameters.
Here's how I see the paradox and it's connection to the known paradoxes of infinitesimal calculus: Here's how the 'normal' time-reversed setup looks like: you have to apply the infinitely precise amount of force so that the starting kinetic energy of the ball is equal to the potential energy at the top of the dome. And apply the force precicely towards the top of the dome. Now, there are two ways to treat that. You can just have this setup in isolation, with that precise amount of force directed precisely where you need it. Or you can recognize that the calculus of infinitesimals, and by extention the Newton's laws, are a product of a limiting procedure. Let's think about the limits. Say the starting kinetic energy is a bit off from the precise number you need. What you see is that the ball climbs up top, has a bit of velocity still and rolls down. Now, the smaller that imprecision is, the smaller the velocity on top will be. And that ball stays on top(in a small region around the top) for longer and longer periods of time. Go to the limit, that period of time goes to infinity. Similar argument for the direction of the force. Say the direction of the force is a bit off. The ball gets close, but not close enough. And then rolls down. The greater the precision, the closer the ball gets to the point at the top. Taking both limits, the ball at the top is moving with a velocity that goes to zero and and passing at a distance to the top that goes to zero, staying in any arbitrary region around the top for a time that goes to infinity. Does that create an unphysical result? Yes. But I see that as a result of taking this limiting procedure too seriously. And in physics, you don't usually treat those like that. Take thermodynamics. You work with infinitesimals, but you want those infinitesimals to still be physically large. Say you take an infinitesimal volume, it should be still big enough to contain lots of particles. Same with electrodynamics. You can work with charge densities and work with infinitesimal volumes, but you know that those infinitesimal volumes are physically large and contain lots of charges. Because infinitesimals are a calculational tool. A tool that tells you the asymptotic behavior. How does stuff behave when you get arbitrarily close to a specific setup? And I think that's the right way to understand Newton's laws. But I thank you and thank the author for bringing this paradox to my attention. Wouldn't have thought about that this deeply otherwise Edit: As correctly pointed out by @deinauge7894 this argument is flawed
your description would be valid for a "normal" dome like a spherical or parabolic one. But this thing is constructed as a rotated r^(3/2) curve. And the rolling time is limited by a finite number - it can not be in a small region near the top as long as you want. (Similar to a cone shape where thats also impossible) Note that the ball also has to be infinitely small for the whole thing to work.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. The apex of the dome is infinitely small or a point which according to the uncertainty principle means the ball always has a non zero momentum -- the flatter the dome the less you know about its initial position. In the real world of quantum mechanics the ball always has a momentum -- heat or uncertainty, errors. Action is dual to reaction -- Sir Isaac newton or the duality of force. The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Symmetry breaking always causes the ball to move -- cause is dual to effect or correlation. Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum duality. Initial or first cause is dual to final cause -- Aristotle. Duality via the uncertainty principle causes the ball to move. "Always two there are" -- Yoda. You cannot have symmetry without anti-symmetry in physics! Duality creates reality.
Yes! Calculus will not always give "correct" results. This type of acceleration is not something that is effectively encapsulated by calculus, so the results obtained merely demonstrate such a result. The claim of spontaneous motion feels similar to claiming that the area underneath a 1/x curve passing through 0 does not converge strictly because the improper integral does not convergence.
@deinauge7894 okay, I think I see what you're saying. My argument totally neglects the fact that a ball that is not at the top is accelerated so although its velocity gets arbitrarily close to zero, it doesn't stay that way. Basically the limiting procedure I describe doesn't even approach this paradoxical behaviour at all. Which to me seems like now the question of whether to accept this paradox as part of Newtonian mechanics boils down to 'Do we accept solutions which rely on exact initial conditions and cannot be obtained by a limiting procedure?"
This is a storm in a teacup! I build pendulum clocks for a hobby. They work because the period for small angles of swing is quite predictable and constant (Galileo). However, the period gets longer and longer as the angle of swing increases. In fact, the period is infinite when the pendulum bob is vertically above the pivot point. You have to look at Gauss' Elliptical Integrals of the Second Kind and use Arithmetic-Geometric Mean to get an accurate answer (Seriously, Gauss was probably the best mathematician EVER!). The pendulum has TWO equilibrium positions - the usual stopped clock position of straight down is a stable equilibrium AND unstable equilibrium position with the pendulum pointing straight up. As mentioned above, the period with the bob straight up is infinite. Suppose the angular displacement from the vertical was 1 femto-second of arc. You'd stand there looking at it for ages (eons even) before you'd notice any displacement from the vertical. OK, in the real world air currents would unbalance the pendulum or static deformation at the crystal level, etc. would unbalance it. The marble on the hill is the same situation. It is not possible to exactly place the marble so that the moments in all directions are exactly balanced. The thing sits there and obeys Newtonian Mechanics. It is just that the imbalance force is mind bogglingly small to begin with so the acceleration is also miniscule to begin with. It is not that one second it is doing nothing and then it decides to move. Here is something worthwhile to ponder: single photon interference made simple. Thanks!
Exactly. The world IS deterministic: and not only that, but it cannot be any other way, because it is based on cause and effect. This video is just misleading clickbait.
Sorry, no, you misunderstood the point of the video. The system you described roughly corresponds to the differential equation x''=x (with x=0 being the topmost position of the pendulum). This *does not* display the behavior discussed in the video, and is thus irrelevant. There is no solution where x=0 for t0 for t>T. If it starts in equilibrium it remains in equilibrium. This is is roughly equivalent to a dome in the shape of a simple quadratic paraboloid. To obtain the phenomenon in question, you need to have x''=x^a where 0
@@apokalypthoapokalypsys9573 The world definitely can be nondeterministic. There is no problem with a universe with clear mathematical cause-and-effect rules, where those rules have randomness built into them. If you think of the universe as modelled by a cellular automaton, for example, no reason the evolution rules of the automaton shouldn't have randomness. As for whether randomness actually exists, this can depend on our interpretation of quantum mechanics; but in practice, the universe definitely is nondeterministic. Decay of radioactive isotopes, for example, is completely random and unpredictable. Bell's theorem shows that the behavior seen in quantum mechanics cannot be explained with a theory of "local hidden variables", that is, an internal state which could be used to predict this behavior if known.
This whole quest seems to stem from a statement that goes unquestioned in the video. "if you accept that it's possible to nudge the ball with just the right amount of force so that is rolls up to the apex and stops there," then by time reversal symmetry, the solution that the ball must at some point roll down with no external force must also be valid. WHY, I'd like to ask, must we accept tht it is possible to nudge the ball just so? If we are going to base our intuition on this, it is worth examining whether it is even valid to find that a ball can come to rest perfectly still atop the dome in an idealized probelm. I suggest that the space of possible "nudges" is incomplete, that there is a pinprick where the possible force vectors would hypothetically lead to the ball resting at the top of the dome. I suggest that this pinprick itself is a disconinuity of sorts, where it's simply not possible. A missing poin, where the limits converge together, but the poin itself doesn't exist. take, for example, "f(x)=x/x". The limit of f as x approaches zero is 1. but x=0 does not have a valid solution. I propose that through this sort of discontinuity, one cannot actually roll a ball such that it comes to perfectly stop at the top of the dome. As the ball goes to the top of the dome, the velocity of the ball approaches a value very close to zero. As the ball rolls down from atop the dome, the velocity of the ball rapidly leaves the vicinity of that same value. The more perfectly one rolls the ball to the top of the dome, the longer the ball spends in the vicinity of the apex. To simplify things, let's consider just the two-dimensional case. A disc is rooled up a curve so that it reaches the apex. Instead of trying to attain the perfect position, let's frame the problem in terms of how long the disc spends at the apex. Since we know that rolling it too hard will cause it to roll over, and too softly will cause it to roll back, let us instead concern ourselves with optimizing for a much more controllable and much less finicky variable than which way it rolls: how long does it take to leave the vicinity of the apex? More specifically, how can we maximize the time it spends in the vicinity of the apex? So, we can experimentally determine the rough magnitude needed to reach the point at which the direction of the disc switches. That's perhaps a good starting point. But as we tweak the exact magnitude, the amount of time spent at the apex can only *approach* infinity. And here is where the crux of the problem lies: We cannot solve for the case where the time spent at the apex is equal to infinity. The function we can construct is necessarily discontinuous. If we could say that the ball could be rolled into such a state of being perfectly balanced, then yes, time symmetry would dictate that a perfectly balanced ball must roll down at some point without external forces. But I argue that we cannot in fact say that the ball can be rolled into that perfect balance.
This is the most succinct description of my problem with this model. There is simply no way to have the ball come to a complete standstill at the peak of the dome, as the peak itself is an undefined reality.
it is correct that if it's based on an assumption that's not true... then we can't guarantee the truth of the conclusion... after all... she did call it a hand-wavy explanation...
@@ntsure2436 But at least the limit of situations that come close to the theoretical limit are not in some far removed infinity, they seem quite normal. But it would need quantum mechanics to describe that case properly - and it gives a quite reasonable answer too.
Actually time reversal symmetry does NOT imply that the ball roles back down at some future time, because in that case time would still be running in the same direction, and is not reversed. Time reversal symmetry just means the equations of motion can predict earlier states given a later state just as well as they predict later states from an earlier (initial) state. Conservation of energy says I can roll a ball so it stops at the peak of the dome. There is no law in classical mechanics that says it ever has to roll back down.
Actually with the specific dome shape in the video the time cannot approach infinity. The ball will either roll perfectly to the top after a finite amount of time (let's say 5 seconds) and then stay there forever, or it will start rolling down again after at most 5 seconds. If the dome had a normal shape, like a half sphere, then your argument would indeed be correct.
19:24 If it is the last instant at which the ball does not move, the causal instinct is about the terminating cause. What made it terminate the motion?
Interesting but I’m wondering if the treatment of time at around 17:25 is partly responsible for the weirdness as normally time would be considered a continuous variable, whereas here it’s treated as though it is discrete. As a result there appears to be a time when the ball is excited that is separate to the instant before when it is still . However if we insist upon time being continuous then the separation between these two snapshots has to be zero as we have infinite resolution on the values of time. This would then mean that these ‘two separate’ snapshots are only in fact one and so there is no moment of excitation to consider and we return to the more physical idea that the ball is either still or moving down the ramp as you’d expect.
Time isn't treated as discrete in the video. There is the last moment when the ball is still (t = T), and at every time after that the ball is rolling. So there no first moment when the ball is moving and there is no gap.
Like others have said on here, the Earth is moving, crust moves, atoms constantly are moving. Also, just the heat from your fingers on the part of the ball where you grab it can cause changes in temperature differences within the ball material itself causing minute shifts. So many factors at play really.
21:00 This is exactly why programming takes ages. Even when we're the one that wrote the exact rules for the computer program, even when the computer follows those rules to a T, even when we didn't want to have a complex idea, there's always more popping out of the woodwork. We don't know the full extent of the flaws or merits in our thoughts, because we only have so much internal RAM to run them.
I read through the math of the original paper. The author bent over backwards to find a solution to the equation that wasn't smooth, and didn't have derivatives with continuous functions. This whole controversy ends when we demand that Newtonian mechanics allows only smooth and continuous solutions.
Physicists literally hate philosophers. Check out the papers on 'supertasks'. One of them concerns the spontaneous generation of movement in a tube full of balls. All that that does is to provide back-up for a perpetual-motion scam. But the difficulty does not have to be very deep. Years ago, in Amer.J.Phys, there was a calculation that 'proved' that a vertically balanced pencil would fall over. Of course it would in real life, but not in theory. The paradox was traced to a subtle point concerning the instantaneous contact-point.
That demand prevents you from being able to solve a bunch of very reasonable problems. For example, suppose that you want to model a car driving off of a cliff. The solutions to simple models of this will be non-smooth as the forces on the car change instantaneously when the car goes over the edge.
@@danielkane8568 Elementary textbooks are full of examples in which vehicles travel for a while at one velocity, then change to a different velocity, and so on. This leads to a sharp-angled distance versus time graph, and students are asked questions about the total distance travelled. I used instead to ask students about the condition of the passengers. They were baffled. Not one ever 'got the point'. The passengers would be dead.
It may also be determined, that most research papers are behind a million pay walls, and thus not available to the public. Yet, most of the said papers, are in large part paid for, buy the very same public.
I read it. And have aerospace engineering degree. I would liked to seen how he derived the solution for the arbitrary T equation as I'm rusty on diff eqns. That being said, could it be that T is infinitely long in the 'real world' ?
@@FrodeBergetonNilsen I just googled it and it is available on the first result. Also, that argument still isn't applicable. It's like leaving a review on a book based off someone's not completely accurate recap of that book just because you can't "easily" access it
The scenario seems to imply that there is a fixed time T at which the ball will roll down the dome. But is it possible to put any bound on what value T can be? And what would influence the value in multiple iterations? T does not seem like it is defined well enough to say anything about a solution involving it.
In ECONomic growth theory, similar laws of motion exist. Here (Newtonian physics), movement leads to slope, which leads to acceleration, which leads to more movement. There (Solow 1956, link below, winning the Nobel price in 1987), capital generates output, which induces savings, leading to investment and thus more capital. Here, you can rewind the time to the state of zero motion. There, you can rewind time to zero capital. Here, you have the excitement time where motion just gets going for no reason. Same there: capital out of nothing. We (Andreas Irmen and myself, Hendrik Hakenes) pointed that out in a research article "Something out of nothing? Neoclassical growth and the 'trivial' steady state" (SSRN 2006, Journal of Macroeconomics 2008). Here: controversy. Our article was cited 7 times (end of 2024), so in economics: not much of a controversy. The story of our paper goes like this: I had studied physics and mathematics, including some differential equations (by the way: in Bonn, where Lipschitz taught from 1864 until 1903). I then moved into economics, taking a growth class with Andreas Irmen. He taught us students the Solow model, and I observed: Wait, this differential equation (the law of motion) does not satisfy Lipschitz continuity if capital is zero. So let's see what that implies... We even had a correspondence with Robert Solow: his reply to our letter hangs in my office, I can see it right now! Links: piketty.pse.ens.fr/les/Solow1956.pdf, papers.ssrn.com/sol3/papers.cfm?abstract_id=892732
@heha1390 1956 I wouldn't think so. But Crypto is a form of capital. It spontaneously came into being. I would say both fits the serious narrative and the joke.
I'm reminded of Zeno's paradoxes, in which an arrow cannot move because at any given instant it is stationary. If time is quantized, then that means there must be at some level a sharp divide between the time when the ball is moving and when it's not moving. Perhaps it is easier to consider the case when it rolls up the dome and stops exactly at the top: what does that look like, in the last analysis? One can only break motion down into pieces so small, before one must fall back on a contest, either- or, scenario. The main problem with Newton's laws is not that they are not deterministic, but that they assume a Continuum.
Calculus does tell us that an infinitely growing sum of infinitely smaller pieces get to a finite result, though. That's Newton's massive breakthrough for me.
Thank you! I know very little of physics but for most of this I was thinking of that thing where the arrow is stationary at every moment of time even though it's very obviously in motion. Thought it was called Time's Arrow but turns out that's a whole different thing. Thanks for pinpointing my place of pedestrian ponderance
How can it matter for a ball if it's on the top of a dome? Would it start to move on a flat surface? On the top of a cone? I don't understand this video.
How can you calculate the exact amount of potential energy you transferred to the ball when you placed it at the apex? In my opinion, it can never be zero. Hence, it is the same potential energy that makes the ball roll down at some point. So how is there a paradox here?
The more common term for things that don't have an instant at which they start moving is 'stationary'. I'm not entirely convinced that stationary objects are moving, whether deterministically or not. If the Big T Excitation time is the bog standard 'somebody pokes it with a finger' sort of excitation, then yeah it starts moving. If Big T Excitation is some unknown indeterminate event where nothing causes it to move but it does so anyway, then it's magical what-iffery, not Physics. If Big T is "we don't know when or how, but at some T in the future it might do" then it looks awfully like a divide-by-zero scenario is approaching.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. The apex of the dome is infinitely small or a point which according to the uncertainty principle means the ball always has a non zero momentum -- the flatter the dome the less you know about its initial position. In the real world of quantum mechanics the ball always has a momentum -- heat or uncertainty, errors. Action is dual to reaction -- Sir Isaac newton or the duality of force. The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Symmetry breaking always causes the ball to move -- cause is dual to effect or correlation. Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum duality. Initial or first cause is dual to final cause -- Aristotle. Duality via the uncertainty principle causes the ball to move. "Always two there are" -- Yoda. You cannot have symmetry without anti-symmetry in physics! Duality creates reality. Idealism (universals) is dual to nominalism.
@@hyperduality2838 as defined at least three times in the start of this video, this video is only talking about Newtonian mechanics and explicitly _not_ including quantum mechanics.
this dome is shaped in such a way that if you roll a ball up the hill, it can completely stop on the apex in a finite amount of time. therefore, the reverse situation where the ball starts rolling *down* the hill in finite time is also allowed in the maths. it isn't pushed down the hill, just like there is no finger to stop it as its rolling up the hill. in the uphill situation, T is the time at which the ball becomes completely stationary. in the downhill it is the time at which the ball is rolling downwards after. The reason this is such a big deal is that the deterministicness of the **theory** of newtonian physics was assumed by a lot of people. theoretical physics as a whole seems like what you would call "magical what-iffery", so in the sense that this only has bearing on the theory, it is indeed magical what-iffery. there aren't any divisions by zero going on, for the record.
@@chartroniumdude5870 "it can completely stop on the apex in a finite amount of time." Actually in mathematical sense -- in an ideal environment of perfectly smooth surfaces and zero friction -- the ball WILL take infinite time to reach the true apex if you give it just enough impetus to reach the true apex. And, that is why the whole discussion is non-sense -- IOW, in the reverse scenario, you need an infinitely small force to get the ball rolling at 'zero' time. Which also solves the 'paradox' or whatever we want to call it: No matter how subtly I move the ball (when trying to place it on the apex), I can't get it there in a finite amount of time, so the ball will always roll off (from the non-apex) in a deterministic way when I release it.
@@_pitako Maybe it should be. Either that or call things that are well established something other than "scientific theories". Perhaps then there wouldn't be people saying things like "gravity is just a theory".
As far as I understand, in physics, had no concept of equality. You can never be exactly at the top of the dome. Any attempt to "idealize" a situation leads to mathematical problems that don't exist in reality.
Awake is dual to asleep -- different states of the same thing, consciousness. Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle. The apex of the dome is infinitely small or a point which according to the uncertainty principle means the ball always has a non zero momentum -- the flatter the dome the less you know about its initial position. In the real world of quantum mechanics the ball always has a momentum -- heat or uncertainty, errors. Action is dual to reaction -- Sir Isaac newton or the duality of force. The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual. Symmetry breaking always causes the ball to move -- cause is dual to effect or correlation. Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum duality. Initial or first cause is dual to final cause -- Aristotle. Duality via the uncertainty principle causes the ball to move. "Always two there are" -- Yoda. You cannot have symmetry without anti-symmetry in physics! Duality creates reality. Idealism (universals) is dual to nominalism.
Not uninteresting (the paper, not the video), but I thought this was long known, and I don't find it shocking at all. The Picard-Lindelöf theorem is usually presented alongside uniqueness violations, one of them has a ball rolling on the edge of a cylinder, perfectly balanced, but which could fall to one side at any point and still form a solution of the equations of motion. The equations of motion are merely constraints over what motions can take place, though generically, they do yield a well-determined ODE.
From classic physics in the real world, not math infinitely smooth world, the ball falls because of deformation of the material of the dome (elastic and plastic), thermal agitation/expansion/contraction and air currents.
THIS! There is motion somewhere that causes the ball to start moving. It’s not spontaneous at all Just because you can or measure it or see it does not mean it does not happen Maybe over the course of the next 20 min this is going to be discussed but 2:41 in it is being proposed that there is some earth shattering concept contrary to this As a philosophical concept like trees falling with no one there making sounds great. From the concept of teaching or learning or making a point…. This seems like the de-evolution of critical thinking in science
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An object at rest will stay at rest insomuchas external forces act upon the object. Get it right. There are always external forces.
@@Alice_Sweicrowe the external forces are balanced in equilibrium when the object is at rest: their vector sum is 0. Jade shows this with g = - N in a diagram.
@@earthpassenger999 Ok, let's see a physical demonstration.
The issue I have with the suggested time reversal or time symmetry proof is that time symmetry is theoretical and that the slope and final location of the ball will only obey that if the top of the dome is sufficiently thin, and it's difficult to know how many atoms can make it sufficiently thick. Besides we need to work at the molecular level because any dome woulds must be stable
@@jermsbestfriend9296 You get it and you've stated the obvious problem and why it doesn't have a chance. Well done.
In the mathematical solution, the dependence on t^4 presents a discontinuity in the 5th derivative of position, meaning that the "acceleration" of the force suddenly changes from 0 to non 0 at the excitation time. This shouldn't make physical sense, in the frame of Newtonian mechanics, because the first law of motion should trace back to all possible causes of motion. Not only should a body experience zero acceleration in the absence of a net external force, but it should also experience zero jerk (change in acceleration) in the absence of a change in the net external force, and so on. Here, we're actually introducing an instantaneous "force" equivalent to the 6th derivative of position, which causes a discontinuity in the 5th derivative, and propagates all the way to the position, appearing to start moving spontaneously.
Therefore, if we restrain Newton's first law to just encompass acceleration, it's easy to destroy determinism, just design a system where acceleration is 0 but its derivatives non 0 (such as the dome). Then you forget what's happening beyond that point. The discontinuity in the derivatives of the acceleration isn't given by the system itself, but it behaves as another initial condition, the one that actually determines the excitation time. It's like saying that you don't know the origin of the changes in the force, and the changes in the changes of the force, then, of course, the system is indeterministic, you are simply lacking information. But if we describe the system as just the dome, there should not be any such discontinuity, thus, no excitation.
Instead, if we include all its derivatives, we must determine the behavior of all forces and changes to these, and the ball will move accordingly, therefore determinism is conserved.
Or NOT move accordingly.
WOW! I think you got it.
Wait, it's the 4th derivative of position which is discontinuous, not the 5th.
@@TH-cam_username_not_found yeah you're right, I miscounted position as the first
All of this takes a funny shade when considering it from a game dev perspective. Gives a good framework for shaping the start and stop times of something's motion, for example.. sentient motion operates on that 6th derivative (or higher) level, it's not just raw changes to position, velocity, or even acceleration. It's changes in rates of changes of rates of changes, ad nauseum.
Hmm. Former theoretical physicist here, and this is the first time I've seen this... but my read (after reading his paper too) is that this is a convincing argument that Newtonian mechanics (and really also Lagrangian or Hamiltonian dynamics, you'd have the same problem) doesn't work in non-Lipschitz-continuous situations. Given the number of other places where it fails (relativistic, QM, etc) that isn't fundamentally shocking, it's more a "good to know - if you're not Lipschitz, you'll have to go beyond any simplified world picture" note. Fascinating.
Also, I like how his example includes spontaneous symmetry breaking, as the descent angle is completely arbitrary. It highlights how once you lose Lipschitz, all determinism goes out the window!
Indeed, this is our first ODE course if calculus of variation techniques cannot help refine the conclusion (surprisingly sometimes it can): Banach leads to both uniqueness and existence, Schauder only leads to existence.
That is also where the "paradox" of breaking of "conservation of momentum" comes in for Brownian motion when we have infinite p-variation when p < 2 (meanwhile one can somehow save determinism by adding more information, e.g. description of higher order processes)
I like this way of viewing it
For me, this is the "solution" of the Big Bang. The Space-Time is the cone and something outside is rolling down on us.
Not a physicist here. Does this article prove that there is the excitation time after which the ball will move? To me it looks like two different scenarios combined to create a false sense of continuity. We can just as well consider a scenario of a ball standing on a flat surface, and after the time T rolling with uniform motion from exactly the same position.
@@YonatanZunger sure but isnt it a relevant question whether (this sort of) non-Lipschitz situation can ever have physical existence? A perfect dome seems impossible to make, and it seems just as impossible to balance a ball on its point
"How was your Christmas?" "It was wonderful. I got fancy slippers and I hate determinism."
that's brilliant, kudos!
Ikr? Like some philosophical idea is going to "break" physics. Smh...
@@geoffshelton2662 I think Jade made it very clear that this thought experiment specifically does _not_ break physics.
@@Kwauhn. Disregarding the title much...?
@@geoffshelton2662 Disregarding the contents of the video much? The title doesn't even say it "breaks physics" lol.
Applying the "exact energy" to flick the ball up to the dome apex results in the ball taking infinite time to reach that apex.
When you try to run this in reverse and realize that the ball can indeterministically choose any path, you have to start at and work backward from T=∞, which you can't do.
Free will and dark matter. Aveo
Exactly. In the described scenario the ball is always moving to the apex, never reaching it. The only way to get the ball at the apex at an X time with horizontal speed = 0 is to place it there.
This would be true of any dome which had a finite radius of curvature at its apex. However the whole point of Norton’s dome is that its radius of curvature is zero at the apex, meaning that a ball rolling up can reach the top in finite time.
@@mikebrown5223 the problem is not with the curvature radius. It's about limits. At the apex the ball has no force being applied to it and no velocity.
However, how does something that is moving stops without a force being applied to it? It can't.
The ball near the apex of the dome is subjected to a force and a speed very very close to zero, but it will never reach exactly zero (only at infinity).
@@mikebrown5223 not without friction, because when it reaches the zero curvature area of the apex with a non-zero velocity, it would continue rolling to the other side with nothing to stop it, no?
If it moves, it's biology, if it smells, it's chemistry, if it doesn't work, it's physics.
And if it's all 3 there's a high probability that it's a teenager!
And if it doesn't exist at all it could be either Trump's brain, not to confuse with a Boltzmann brain, this entity COULD actually exist, or a "metaphysics" course sense in a Spanish speaking university
@@swanronson173 Most teenagers I know don't like to move very much. :D
Hence it's all physics.
And if you don't understand anything, that's philosophy.
"physics care about math, but math does not care about physics-" this is such an awesome statement!!!!!
But why we do act like physics is a unique scientific field on regards to math. All fields, especially concerned with precision, use mathematics.
Actually the reverse is now the case, physics has kicked math out the door, and now believes in random unmeasurableness
Physics = men
Math = women
@@achillesglacia7700 Current academic physics does care a lot about abusing mathematics for it's weird dogmatic reasons. Wolfram's new paradigm is a different case, and there are also other constructivist pure mathematicians who contribute also to physics, Louis H Kauffman, Norman Wildberger etc.
Mathematics, both applied and pure, are not a scientific field@@ThatTimeTheThingHappened. Mathematics is more closely related to a blend of language and music.
In the dome paradox, the "last instant of rest" is like the supremum of a set: it marks the boundary between rest and motion. However, there is no "first instant of motion" because motion begins in an open interval where any candidate for the first moment can always be preceded by an earlier one. This is similar to how a set like {x > 0} has no minimum, even though it approaches zero. Thus, the transition from rest to motion lacks a specific starting point.
Perhaps this all condenses down to quantum mechanics whereby the ball and dome are not truly completely at rest with respect to one another (and to the rest of the universe), but all it would take is some random effect to start the ball moving discernibly.
@@mauricegold9377 what you’re saying is true, but outside the scope of Newtonian physics. Otherwise just by having heat, the ball would move enough to move its centre of gravity out of equilibrium. Newtonian physics is a simplification of all of these things which is why everything is a ball with a net force.
are we simply observing the ball at rest because the acceleration and velocity are so close to zero that we are unable to see it?
So the explanation at 22:58 said, that there is no smallest positive (
I agree... In my opinion, continuity (and therefore its siblings like uncountable infinities) simply messes with our logic and intuition. This example of "there is no first moment of motion" is yet another delightful example of these alien realities we only discovered and tamed thanks to mathematics. I think our ancient philosopher friend Zeno would have hated this ^^
To the people claiming that non-smoothness or inexistence of snap at t = T makes the example unphysical, while I don't see Newton's laws prohibiting it, I will instead change the example to a smooth one:
Note that in the paper r is distance along the surface of the dome, not just the horizontal distance, thus v = dr/dt.
h is the loss of height so it will be positive.
Use r(t) = e^(-1/(t-T)) for t > T and r(t) = 0 for f 0 and h(0) = 0.
This follows from mgh = 1/2mv^2.
The argument that there is no first instant of motion and threfore no need to seek a cause has real Xeno vibes!
i know. this is what happens when philosophers get involved. I can't show that mathematically it is wrong, but it doesn't make sense mentally.
We tried to shoot that idea down but it moved before the arrow could hit it.
Yes, but from a mathematical standpoint makes perfect sense.
It's the same question as whether 0.999... == 1.
There’s something wrong with infinity, either the concept itself or our understanding of it.
1:19 Honestly the most shocking revaluation to me in this video was being told 2008 was 16 years ago 💀
Jesus christ
I feel like it should be more clear that this argument is not saying balls can spontaneously start rolling. It's saying that Newton's laws fail to describe some edge cases. We know that the ball will not start rolling, but we cannot disprove it with Newton's laws. This is just an issue with the framework.
Let me reword this due to comments: we know that within Newton's framework, balls are not supposed to spontaneously roll. What's being argued here is that under the same framework, balls can technically start rolling spontaneously in certain edge cases. If that's actually true, there is a flaw in the framework. This really has nothing to do with the real physical world.
Reword #2: it's pretty clear Newton's model was intended to give a deterministic prediction given an initial state. If it fails to do so in some situations then the model is flawed. The model is not attempting to account for any real world phenomenon that might cause an indeterminate solution.
Newton doesn’t have laws - it has decent approximations for everyday use. This is the issue calling them “laws”.
The physical realm is indeterministic. The maths do not figure into their equations the physical aspects of matter due to its varying energetic states and pliant tendencies.
I think you haven't understood the point of 16:10-16:35. Your statement "We know that the ball will not start rolling" doesn't make sense - it sounds like you're trying to describe what would happen if we set up the system perfectly in the real world, but it is not possible to set up the system perfectly in the real world (because, among other things, physical matter is not an actual continuum).
@@DrJulianNewmansChannel In that case is it really a problem with the theory? Newton's laws describe how things behave in reality, they aren't prescriptive, they are descriptive. Should we be surprised that they fail to describe situations that are physically impossible? The laws describe reality, not prescribe how things should behave in a non realistic scenario.
I'll understand the 'argument' better when someone explains how she suddenly pastes the term "excitation time" into her spiel without definition and we don't see it again until some 9 minutes later:
"...or as Norton calls it, "before the excitation time"
without substantive explanation; I'm not interested in Norton's "before the" part. I appreciate the fact that she pulled it out of his paper, but given the length of this video and wealth of preliminary crud in the first 4 minutes, I think a little more thought could have been given to a complete discussion and development of the concept, as the whole argument seems to balance precariously on its dome.
Nobody in the comments seems to think the thing needs any _a priori_ development either. They just plug it into their own arguments like a well-understood given.
I'm struggling with the idea of T (excitation time)... if the original conditions was a ball placed on the apex of a dome with no external interference, then (in my feeble eyes) the implicit nature of such an excitation time seems to imply that indeed something does effect our ball. Also, the deterministic answer is r(t)=0, but for what interval? I am guessing it would be 0
If you look at the mechanics of deformable solids, you end up with lots of non-deterministic situations like this. For example a truss member loaded in compression will buckle when the load hits a critical value, but the direction in which the member will buckle is non-deterministic. To solve such problems, engineers often have do bias their numerical simulations slightly to force a given bucking direction so that the simulation can run without crashing.
Is this non-deterministic or rather you lack all the information to determine it? Like internal structure, steel crystalline lattice of this specific truss etc which makes it buckle this or another way.
From a programming point of view it should be easy to assign a random buckling direction. Just saying.
@@numbersix8919 that has zero predictive value
As a structure engineer, we describe buckling as a harmonic of nth order, where n is the number (how much) the system’s statically indeterminate. (For linear elements with constant cross section)
As far as i know, programs output all the possible buckling outcomes, however, you are right, in order to force a ubiquitous bucking direction, we must “lean” differentially little on one side.
Dominion (2018)
5:00 Why should I accept that it's possible to nudge the ball exactly the right amount to get it to the top? I feel like it's weird that this is just getting glossed over, and I'm hoping this gets addressed later in the video, because my intuition is that you always arrive *at* the apex with a nonzero velocity (which you need to move from one place to another) and that the velocity you came in with will *always* carry you over, if you make it to the top. You can get infinitesimally close, but not quite actually to the exact point.
I think that is the critical flaw in this paradox. It's _not_ physically or mathematically possible for the ball to stop moving with zero acceleration, and at the exact top of the dome, there is zero acceleration (else it would be impossible to balance the ball there in the first place).
Unless there is friction, it is physically and mathematically impossible for a rolling ball to stop rolling at a point where it could theoretically be balanced. Balancing the ball requires zero acceleration. Stopping it requires nonzero acceleration. The same point cannot fulfill both conditions.
having finished the video, I remain unswayed. I'm convinced this is a modern reinterpretation of Zeno's Paradox and nothing else.
I have the same question. The whole video based on false statement
If you made a mathematically perfect version of this dome and give the ball a push perfectly upwards with the perfect velocity, then it will indeed come to rest at the top indefinitely within a finite time.
(Or at least that outcome would be completely consistent with Newtonian mechanics)
I think, for me, the idea of "infinitesimally close" is at the heart of all this. There is a basic assumption that there IS (or should be) a point where the dome is effectively horizontal and the ball can rest there. In fact I think that may be impossible to reach, not just in reality, but in the maths as well. You can never actually be in equilibrium here, you can only ever be infinitesimally close to it, hence there is no actual equilibrium. It's bothersome to the human mind but I think it may be true.
I'm not sure I buy the argument, but it did get me thinking about trying to push the ball up the hill at such an exact speed that it will stop there, assuming a completely smooth surface of the dome and the ball, i.e. atoms aren't a thing. If the speed was off by just 0.000000000000001 m/s (or less) the ball would appear to stop for a period of time, and then begin rolling again. The smaller the error, the longer the ball will appear motionless (but is actually moving infinitessimally). I know this isn't the thought experiment discussed, but it's still interesting nonetheless.
I was think along the same line. If we could put the ball arbitrarily close to the center, then the ball could seem motionless for arbitrary amount of time before rolling off.
It is still theoretically possible to get the exact speed up hill so that the ball stops at the apex. and there are multiple paths and time this could happen. If we accept that Newtonian physics is time reversal, then it becomes nondeterministic.
I first thought it wouldn't stop at a finite time. Because at that moment i thought it was a normal spherical dome.
But the shape is chosen in a way, that with the exact starting condition it will stop after a finite time. And every other path is limited by that time too! On this dome the ball can never appear to stand still. The only way it can stay up longer is if it rests exactly on the top for a while.
E.g. if you start at r=9, The ball reaches the top at t=6. And it is back at the starting point at t=12 if it starts rolling immediately.
If you get the starting speed slightly wrong, the ball is back at the bottom after less than 12 time units. No matter how close you get to the exact value.
i think in ideal world where we can be perfectly align it, even if u take the side it doesn't role on its own u would also have to admit that such movement is no longer time-reversal.
for me i feel time-reversal is much important core aspect of newton physics then determinism, being time-reversal lead u to think its deterministic which make sense, but here we have example where time-reversal isn't deterministic which disprove our claim that time-reversal mean deterministic.
@@FernTheRobotyou just made a great summary of the whole thing. I think 🤔
If it were possible to place the ball perfectly at the top, mathematics says there’s only one way to do it. Which is kinda surprising
0:45 gave me massive vsauce vibes
Michael isn't here!
If I'm at the North Pole, which direction do I go in to head south? If I'm even a millimetre off, there is only one south. If I'm precisely at the North Pole, every direction is south, so I sit there forever, frozen by indecision (and the cold weather...)
You would not freeze at the North Pole. Santa could make you a cup of tea.,
@@chriscurtain1816 he got shot down
@chemplay866with facts and logic. Get rekt nerd
Sorry for your indecision . . . Really this is a case of if every answer is an acceptable answer. Down select if you prefer walking or boat, and start moving toward your goal. Still can not decide throw a pencil in the air and start moving in the direction the penciled lands pointing. Usually "good enough” is "good enough".
The siting frozen is the 1 decision possible that will prevent accomplishing the south pole goal.
And once you get to the south pole, make a snow angle, so you can be sure the point that is the exact south pole, will be withing a 30cm x 30cm box that is your chest.
The sun is always south at the north pole, so you just have to head towards the sun.
0:50 "Hey guys, Spirit of the Law here."
Extremely specific comment but I’m here for it
11
Shout-out to SquareSpace, for sponsoring my new business: Spirit of the Dome.
@@jamieclarke321 its the background music used that resembles another's youtuber usual use of background music, that of Spirit of the Law, a mathematician who analyses the Age of Empires II game and the franchise.
I'm going to talk about the difference between cataphrats vs. samurai in the castle age and imperial age.
There is a divergence at r=0. It means the description at this point is unphysical. Nothing to philosophize about it, in my opinion.
It is somewhat similar to physics of shock waves. In the past, people thought there cannot be a sudden step jump in physical parameters at the front of a shock wave. Which is true. Therefore, many considered the shock wave itself as non-physical. Later, people learned that the change at the front is gradual on the molecular scale, which seems sudden on the macroscopic scale. So no problem there. Today the theory of shock waves seems well established.
They said that Newton is non deterministic. But they based it off of a velocity equation that was non-deterministic. So you can't introduce non-determism and then say well this must be Newton's fault and not my own.
What do you mean "a divergence"? The hill is continuous (and in fact differentiable) at r=0.
@@danielkane8568 the acceleration is equal to square root of r, which diverges at r=0.
Having a divergence isn't unphysical. General relativity has divergences all the time that disappear by coordinate transformation. In this case, you describe the acceleration in terms of the distance, but why not in terms of the time (in which case it disappears)?
@@duncanhw I think that generally in physics a divergence means that some other physics is missing that is smoothing the effect. For example, if I remember correctly in GR there are divergences in black holes that are smoothed by quantum effects.
Thanks for pointing this interesting problem! As a theoretical physicist, I wanted to better understand, so I did some analytical calculations that gave me another picture of this problem. First, to have a better grasp, it's good to consider all kinds of domes (2d cross-sections are sufficient) with different shapes, and more specifically around the top which is where the problem resides. For usual domes, where h(r) is at least a C2 function around r = 0, one finds the usual behaviour: a ball at rest at the top stays there, because of such initial conditions that determine the future of its behaviour. When this condition is broken -which is the case for Norton's dome but also for a whole family of domes, it can be shown that the same kind of trajectories as those found by Norton exist when the ball is at rest at the top. The indeterminism is not shocking me more than the situation of buckling, where the moment of buckling is not specified by the equations: this would mean that something is lacking in the problem formulation to determine T. What was not considered up to now is that such domes have an infinite curvature (or zero radius of curvature) at their top, which explains why a movement can follow rest at the top, specified by the usual initial conditions. This, for me, is the main surprise of this problem, where such situations are not usually considered.
Experimentally, this is difficult to really observe this phenomenon as a finite ball, friction, perturbations, and precise initial conditions are all a problem compared to the theory. However, one interesting thing coud be to try, for equal conditions, several types of domes to see how the behaviour changes with the dome shape. Answer me if interested!
@ernesthector2451 I think the thought experiment makes no sense in theory at all. The only flat surface mathematically is a single point considered 0. Every ball that exists is bigger than 0, even if just an attometer. It, therefore, would be subject to curvature. Even if just millions of zeros behind the komma. It is just a matter of time until we can perceive movement, but it would happen at the very moment the ball is placed on the apex.
Genuine question here...
I don't understand why the solution to the dome has two possibilities and one of them includes an excitation time.
Shouldn't there be only one solution? And if you place the ball at the top of the dome it will just not move? Ever. I mean, if we were REALLY in the vacuum of space and no forces acting on it other than gravity towards the base of the dome.
If there is an "excitation" at all, shouldn't the excitation be external? Again following all the laws?
In real life if the ball stays still for a moment and then starts rolling by itself, my intuition would tell me that there was an external force. Wind? Small vibration imperceptible to us? Anything that we can't see cause it's too small or weak, but at microscopic levels, it happened. And it's weak but strong enough to remove the ball from its unstable equilibrium point. Gravity does the rest.
So the bottom line is... I don't understand why there's a solution at all where the ball moves spontaneously.
Can someone explain?
That's what I was thinking
@sabapc81 SO... There are many problems in what you have said. You are hardheadedly applying an intuition that has been called into question in this very video. The intuition that "on a ball sitting on the top of a dome will not move".
The real life argument is again a problem: real life follows a mix of relativity and quantum mechanics. Even better: we have no idea what real life follows and quantum mechanics and relativity are our best approximations.
You are applying a model a little bit hardly on to a reality we do not know how behaves at plank's distances.
The key point is 3:10 this is a model, and we are working with the model and testing its limits. The second key point is 4:41 where she states time-reversal symmetry and thanks to which if you accept that you can nudge a ball on top of a dome then you get that the reverse is a solution.
There are already two solutions here: the solution of the ball that has been sitting on the top of the dome since the dawn of time and for the rest of eternity and the solution of you nudging impeccably the ball onto the peak of the dome.
This is the precise key point because if the dome was perfectly SMOOTH then the resulting vector field for you differential equation would be smooth or at least Lipschitz continuous and then you actually cannot nudge a ball on top of it. No amount of energy would allow you perfectly put it on top: you either undershoot it or you overshoot it.
Which now that I think about it may also be counter intuitive: you cannot nudge a ball to stay perfectly on top of a dome in smooth conditions. Consequence? Uniqueness of solutions.
Excitation is just a funny name: there are no external forces.
This is just math.
I am trying to stress over it again and again because it is not clear to me which is the problem and what is exactly the point that should be addressed.
You have said "In real life if the ball stays still for a moment and then starts rolling by itself, my intuition would tell me that there was an external force. Wind? Small vibration imperceptible to us? Anything that we can't see cause it's too small or weak, but at microscopic levels, it happened. And it's weak but strong enough to remove the ball from its unstable equilibrium point. Gravity does the rest."
What I see here is "But in real life the models tells us this" and it is good when you test the model but at the moment you are sticking to the model like a religion. "It has moved, the theory that I have been taught tells me that a force was applied. Where is that force?"
What your brain is grasping at is a mathematical model that we have glued on top of newtonian mechanics for years.
Forces, energies, momenta, potentials are all part of the model, a "smooth" model.
All concepts make sense when all the quantity involved are smooth.
It the forces are only continuous then we have what this video is all about.
Peano theorem grants us existence and not uniqueness for only continuous vector fields.
The video is meant to break your intuition because your intuition was wrong, like ours. Welcome to the club.
You firmly believe that the ball won't move.
The way we have modelled the theory says it can.
We have no experiment to make that can disprove that this case can or can't happen.
Either our way of modelling newton mechanics is wrong and all the concepts defined there should be reworked to accommodate this example, or newtonian mechanics needs to have an assumption of smoothness to properly function.
I hope I made something a little bit clearer and this was not just a random rambling of sorts. I apologise, it is 4:30 in the morning.
@lolmanthecat Thanks for the effort.in explaining. I will watch the video a couple more times. Not quite understood but I won't into it further. It's late here too.
Your intuition is that in the absence of an excitation, an object at rest will stay at rest. But this implies that an object in motion, under these conditions and subject to only constant forces, must always be in motion. This implication is plainly contradicted by models of straightforward situations using standard physics rules. According to the paper, you can use a purely deterministic set of rules to create a nondeterministic result, which contradicts your intuition.
Imagine going all the way back to the beginning of time, or what Aristoles called the first cause, an unmoved mover. How did this excitation happen without anything else? The problem lies within our intuition that the math here tries to counter. So yes as absurd and abstract as it might be, math is still math and it doesn't have to make intuitive sense, but it can bring us somewhat closer to an indetermistic answer.
I am an engineer, I am not so heavily math inclined anymore but two of my favourite courses, gave me memories that are somewhat aligned with what this video discusses.
1. Buckling where a beam was assumed to have a force perfectly acting along its axis, compressing it and we induced a small pertrubation to make it buckle, this pertrubation was explained to us in the fact that no force is ever perfectly vertical and even if it is, there is no beam that is perfectly symmetric and no material that is as perfect as we imagine while studying.
2. Oscillations, where the first thing the professor said is that we are never solving a physical system, but rather a model that tries to reflect the behavior of said physical system, if the results are bad, the model is bad for that specific physical system, but not necessarily for any other. This is why for simple problems we can ignore air resistance, but if you do that when studying the motion of a plane, or a fast moving car, your results are very, very wrong.
In the dome's case, in the real world, the buckling arguments apply there always might be a small pertrubation, from the wind, or from the geometry of the dome itself not being perfect that will eventually cause the ball to roll down.
In the idealized world of math, both solutions are valid, the stationary and the rolling one, but Newton's laws cannot be used to explain the transition phase because they are a mathematical model and the function that describes the ball's is simply an edge case that cannot be so elegantly explained due to the way those laws are built.
I am looking at this argument completely cynically since it is my job to identify areas when a particular model is applicable, when not and why, admittedly taking Newtonian physics as the foundation of everything, but I can perfectly imagine an external factor, one we know exists, or existed, but are not worrying ourselves about how or why, being responsible for the definition of the excitation time T being an arbitrary value, just so the model can make mathematical sense, just like how a small pertrubation causes a beam to buckle, one that can be explained in the physical world, but we do not concern ourselves by defining further, just so we can get the solution we are looking for.
So possibly the idealized world of Norton's dome is perfect compared to our world, but just imperfect enough to allow for this small nudge the ball needs, in a way we do not care to explain where it came from, so we just reduce it to a time we arbitrarily call T. No matter how ideal Norton defined this world to be, the existence of that time T is based on that imperfection, at least in my eyes.
"I don't know how to answer this question but I made up a different easier question in my head and I figured that one out"
@@comedyman4896wow, that's super helpful and insightful of you. Do you have any other Marvels of your intellect to drop on us? Is it commonly held piece of wisdom that if you don't have anything helpful to say, you should just keep your mouth shut. You aren't impressing anyone
@@comedyman4896it does answer the question to me. The dome equation says the ball "could start moving" but it's not proven and cannot be disproven. The answer is we don't really care because in real life it will eventually start moving and we know that, at which point Newtonian physics take over. I'm no expert but when looking at equations that are so large that the dome (paradox?) could add to a significant error if not accounted for, aren't other assumptions that are typically taken with a Newtonian approach likely to generate a far bigger error?
Assuming that the ball and dome are perfect, assume that the ball is perfectly placed, and it is in a vacuum and positioning such that there is no vibration to affect the ball. Please explain how the earth's rotation will not act upon the ball
@@sordidknifeparty Why don't you apply your own advice to yourself? You didn't even understand the point the other person was making, you certainly aren't impressing anyone.
usually if something breaks physics it's because it starts with flawed assumptions or flawed equations, at 18:33 is the possible choking point for flawed assumptions/equations, maybe it needs to be approached from a different angle
If you're also thinking it's being raised to zero, it's not. It's not ^(T-T), but *(T-T). Just skewed writing.
I know right, isn't when t < T then acceleration is negative so the ball is infinitely slowly slowing down right up until it stops and then infinitely slowly speeds up. When it is at actual zero depends on how closely you look at time. Like what is the acceleration of the ball that hits the wall at the exact moment it hits the wall? infinite for an infinitely small time. It is just a break down of Newton at small times and scales and we know it breaks down on that scale already because of Quantum. I think the "special" solution is just kicking a ball up a hill so it stops at the top and then falls back down. With an infinitely perfect ball and hill there is no point in time when the ball stops unless time tends to infinity small. So the ball is slowing down or it has always been there not that deep.
Some possibilities:
What are the elasticity properties of the materials involved?
Is the ball absolutely stationary at the start? Not just "as far as I can see!"
Is the experimental set-up totally insulated from external vibrations, gravitational attractions from the operator,
Etc...etc
"THIS BREAKS PHYSICS" in the splash image is just clickbait, by this channel. It's just more damnable enshittification by the TH-cam algorithm.
I was looking at 17:00 the problem is reduced from motion over time to acceleration during an instant. IMO, if you speak about instants you discard time. It's like a function can be discontinuous in a point and still call it continuous because the limits on each side are the same.
It's always good to learn about the places our present understanding of math breaks down. The more we learn, the more we find there is to learn.
I am at 1:30 in the video, my guess is that the complaint has to do with non-uniqueness of differential equations, here there's a failure of Picard Lindelof theorem induced by the singular point of the dome.
Yes, how many clicks can be squeezed out of an infinitesimal? Something only a philosopher or a youtube creator would ask.
It's that plus the fact that after he concludes a non-falsifiable statement from this, we have hard proof and tons of papers that "no one has ever disproven it" 🤣
I am 4 minutes into the video, I'm convinced all of you are witches and you're clearly speaking in tongues.
4:18 _"That's the point of the dome"_ , while pointing out the point of the dome...
Brilliant 😂
One thing I love about this is it puts “quantum weirdness” (and the competing interpretations of quantum theory) into perspective: it’s not that Newtonian mechanics is “intuitive” and quantum mechanics is “unintuitive”: it’s that Newtonian mechanics is intuitive *enough of the time* to put us off the scent of the ways in which it is unintuitive
Great assessment
No. The classical case is fundamentally different, because you need unrealistic fractional power force laws to get these types of non-deterministic behavior. The person who discovered this is James Clerk Maxwell, in the 1870s, his example is a particle which feels a force which goes like x^(1/3). This is exactly the same as the "Norton hill", so Norton is committing plagiarism, as usual for philosophy.
@@annaclarafenyo8185 All you actually wrote here is that non-deterministic results are “unrealistic”, and that’s simply not the case.
@@annacoeptis It's definitely the case. There is no example in nature of a non-Lipschitz force-law of this type, nor can you construct it on a microscopic level. Further, in quantum mechanics, there is no more ambiguity in these types of force laws than in regular force laws, they are both the same. This example is out of date and also not due to Norton, so it's plagiarism, and not even clever plagiarism at that.
@@annaclarafenyo8185 I’d have to look deeper into it. You know more about the subject than I do, so I’ll take your word for it for now.
the first quarter of this video asks all the right questions in order to be a very “self conscious narrative”, i mean, it feels like a near perfect beginning of a topic. very well thought and executed, kudos.
Newton did not take these types of edge cases into account. The theory is completed by calling that all derivatives of the velocity be equal to zero if the external forces are zero. In this case, the derivative of the acceleration or second derivative of the velocity is not zero for t>T. Therefore the ball does not stay still, Newton's law is not broken, but this implies that there is a disturbance at some instant T, which makes the second derivative of the velocity different from zero, leaving the rest case. Therefore the ball does not move spontaneously. Sorry for my bad english.
Newton stated the 1st law, because it is independent from the 2nd. That's why there are three of them and not two. The example in the video basically shows that Newton's theory is incomplete without 1st law, since the equations of motion from the 2nd law give multiple solutions. It's just so happens that in many problems only the 2nd law is enough.
This is not sufficient because for Real functions, smooth is not necessarily analytic. These exceptions are called flat functions like e^-1/x^2
i believe that is the reasoning behind "we need a 4th law". if the net external forces are zero, all derivatives of velocity must be zero. but once we do this, then we can ask what violations of this 4th law would look like and create experiments that would indicate 4th law violations.
what you're saying is... the 3rd derivative (called the *jerk* ...lol) is not zero... so eventually the acceleration changes from a zero to a non-zero value...
that makes a lot of sense to me... but i think that has something to do with the reasoning behind needing a fourth law
this is also exactly how I think about it. I posted similar reaction separately yesterday, just saw your remark. only difference is that I started with the second derivative of the accelaration as that goes from 0 to 1/6 at the moment the movement starts. so that should be the rootcause.
Great video, but can someone help me out? At 18:33, she equates t and T, which means the power is 0, and then she says "the whole *thing* is 0" - surely it's 1? What am I missing?
It's not written clearly. There is only one exponent. It's supposed to be (1/12)*((t-T)^2) not (1/12)^((t-T)^2).
@steviestickman680 Right, thanks. If there was a derivation of the formula, I might have spotted that - on the other hand, it might have gone over my head and I'd have not finished the vid.
Was looking for this comment, thanks stevie
Was looking for this too, thanks.
18:25 shows the printed form of the formula.
I see the problem: 3:40 your cow isn't spherical.
Of course it's spherical. All cows are spherical. It is in the nature of cows to be spherical.
Yes@@Qarl23
But if it were, how does it interact with an infinitely long one dimensional chicken?
you missed it: it's spherical with radius of zero (aka "the best kind of spherical")
It is not a fat cow.. 🐄
That made much more sense than it should... I mean, I do not subscribe to determinism, neither physical nor philosophical, but it is counter intuitive to think about Newtown's laws in an indeterministic way! Very cool!
one can argue this solution is not physical using an argument from calculus. while location, velocity and acceleration are all continuous functions, the derivitive of acceleration, also known as Jerk, is 0 when t
I don't agree that for physicality the acceleration or even less the jerk needs to be continuous. Consider a block sliding and coming to rest under simple dynamic friction. That example has a discontinuous acceleration, and a Dirac delta -like jerk. I think the conclusion from the video is that Newtonian mechanics simply is not always deterministic.
@@ahhuhtal Well ya don't hafta be a jerk about it, Geez!
Do we actually have reason to believe jerk is smooth always? This is something I have wondered a long time. Certainly solutions are not analytic since at some point objects sometimes stop.
But doesn't that argument fall apart for the time reversed situation out the ball rolling up the dome and coming to rest at the top. No-one has any argument with that being a valid solution of Newtonian equations of motion, but it also has a discontinuity in jerk and divergence to infinity at the point in time at which the ball comes to rest.
@@ahhuhtal Well, we *do* know that "simple dynamic friction" is an approximation for a perfectly-rigid body - which, of course, is untrue. At lower jerks, we know 'softer' things (e.g. tires) have additional behaviour beyond that, which smooth out these transitions by deformation. It should be the case that at larger jerks, even the most rigid bodies in reality will deform.
Take the idea that rigid bodies transfer forces instantaneously. In reality, they transfer at approximately the compressional velocity of sound in the material. It's just that for a sufficiently rigid body dealing with forces on larger time-scales, this is a fine approximation, but if you start pushing at the edges of this theory, it doesn't truly work.
I've never made a TH-cam video for the general public, but I can see that an in-depth explanatory video like Jade Tan-Holmes (of Up and Atom) makes requires an enormous amount of work and research. Yet she is always so fresh.
Indeed; the videos are fresh, energized, and intriguing to curious minds. Rock on, Jade!
So Fresh, Jade has that MOSCHINO vibe 😘👌
Edit : 2025 for 1M sub
She's fresh, she's so fresh, exciting!
She is hot, so hot. Beautiful
She did NOT do her research! If she had done her research, she would have discovered this is a 19th century argument by James Clerk Maxwell to explain the "Lucretius Swerve" of ancient poetry. Norton is plagiarizing the argument, gambling on the public being too ill-read to find the original source.
Not-a-physicist here: From my understanding, I don't see where that breaks "symmetry". Newton laws still apply if running things in reverse, at the very least with a proper value for big T. If you were to perfectly push a perfect ball on a perfect dome so that it stops perfectly on the apex, it would stay there for an infinite time. In reverse (or forward, if pushing the ball was the reverse) the postulated position equations (for t = T) are still valid as long as T is infinity as well.
I like how all this intuition is time-invariant, that any insight here will also apply in reverse. It makes much more sense to understand the excitation moment as the ball coming to a rest at the top of the dome when its kinetic energy becomes zero, and how our understanding of acceleration at that time approaches nonsensical. It makes sense that there would be no "last instant" of the ball accelerating to the top of the dome - this is just another examples of the Achilles paradox.
Another great piece of work. Thank you. The edge conditions that break a model are very useful for at least 2 reasons. First, they show where the model can be improved by understanding the underlying reality better. Improvement on models leads to new physics which leads to new models which leads to new physics... Second, at these boundary conditions where nature doesn't normally go, we find very useful and non-intuitive uses of physics to invent things we never thought possible before.
Particle physicist here: lovely! Very nice. I love dropping by your station from time to time to see what's new. Always very curious,novel stuff, competently and lovingly presented. best regards, DKB
Not quite sure if it's the same problem as the one I'm thinking of, but a similar thing can happen in stress buckling of a cylinder under pressure. For an ideal cylinder (e.g. a soda can) it is impossible to predict where it will buckle, since the initial conditions needed for a prediction are the same all along the radius. It's an interesting quirk of simulations that they typically terminate before buckling occurs, since the solution is indeterminate and breaks the linearity assumptions baked into most FEM programs.
it is perfectly possible to guess where it will buckle in the real world though from inspection of the metallic grain structure. The real world does not have truly ideal cylinders that are perfectly strong to resist compression forces in all directions perfectly, so it's an irrelevant and useless thought experiment.
The universe isn't made up of mathematically perfect surfaces.
That is not analogous to this, its comparable to how Newtonian mechanics will say that a ball at the top of a hill wont roll down, but irl it will roll in some seemingly arbitrary direction, but this is actually due to it not actually being exactly on infinitesimal top of the hill and a variety of other factors like temperature, wind, and things having non smooth atomic structure. From a theoretical pov, the ideal cylinder should never buckle, just as from a theoretical pov, an ideal stationary ball on the top of a hill should never roll. The difference here being that there is a challenge to that theoretical conclusion.
8:25 Your impersonation of me watching the beginning of most your videos before you break things down it's spot on!
I've probably said it before but its worth saying again, your ability to break these highly complex topics down for us in a way that's as in depth as it is entertaining is truly a gift and I'm grateful that you choose to share that gift with us!
If you are ignoring friction, then any speed that can reach the apex will no longer have any resistance to its motion as it moves to the horizontal instance of the apex and so will continue to the opposing slope. If it had less energy it would not reach the apex, there is no perfect speed as to transition from the slope to the apex is to have some remaining speed which is no longer resisted, so it is impossible to land perfectly at the top of the dome though the process described. to be otherwise is to introduce some mechanism that would in reverse explain the behaviour.
Yeah, the real paradox here is the idea that a rolling ball can stop rolling with zero acceleration. That violates Newton's first law.
The acceleration due to gravity goes to zero as the slope of the dome goes to zero. It is mathematically and physically impossible for a ball to stop rolling at the precise point where it has zero acceleration.
@@FirstLast-gw5mg Since there's no friction calling it a 'rolling' ball introduces an unneeded and obfuscating parameter. It's just a 'moving' ball. And it doesn't even need to be a ball, it could be a mummified toad.
@@paulkline515 You're mostly not wrong, but I'd clarify that it can be any object that has exactly 1 point of contact with the surface.
If the object has multiple points of contact with the surface, I think it could possibly end up saddling over the top.
You roll the ball up the dome such that the initial kinetic energy is equal to the potential energy the ball would have at the top of the dome. Newton's laws don't forbid this.
@@FirstLast-gw5mgit is certainly possible, theoretically speaking, to roll a ball up a frictionless hill and have it stop perfectly at the apex with gravity being the opposing force. Looking at just the mathematical model, it just means there is only one solution at any given approach angle that allows for this.
We have to ask ourselves what the first law of Newton actually says. At 17:19 Norton reformulates the first law so that his proposed solution of the e.o.m. doesn't violate it. The problem is that his rephrasing distorts the actual meaning of the first law. Basically, it says a(t)=0 for F(t)=0. But isn't this just a special case of Newton's second law, which says F=ma ? Then the first law would be just a special case of the second law and therefore completely redundant? Was the great Isaac Newton really that sloppy? Short answer: No. The first law must say something different. Let's look at it again: "Every object perseveres in its state of rest, or of uniform motion in a right line, except insofar as it is compelled to change that state by forces impressed thereon." (source: en.m.wikipedia.org/wiki/Newton%27s_laws_of_motion) And now apply it to the dome: Assume a ball to be perfectly on top of the dome. Unless it is slightly dislocated by a perturbation - which we don't assume to be the case - the net force on the ball is zero. Therefore it stays at rest. If the ball is dislocated from the equilibrium point (i.e. the top of the dome) then of course there is gravity exerting a force on the ball which causes it to move. But what Norton doesn't get here is that Newton's first law prevents such a dislocation from the equilibrium point. Therefore Norton's proposed solution violates Newton's first law.
So then what about the time reversal? Does the time reversed solution violate this, and if not then why can we not time reverse?
So, write that paper.
You act like the laws of motion aren’t redundant, but really they all are just simplifications of special cases of the second law. If force is zero an object will not accelerate. That is the first law in terms of the second.
There’s another example though. The third law states that forces are equal and opposite. This is just the first law if you think about it though, because if you take an isolated closed system with only two objects that are exerting forces on each other, the forces must be equal and opposite, otherwise their center of mass would accelerate, violating the first law, which is also violating the second law, since there is no external force on the system, and therefore no acceleration of the system.
The first and second laws can both be completely derived from the second law.
The dislocation starts as an infinitely small dislocation. Since you can flick the ball up the dome and it will come to rest at the summit, you can always time reverse the equations. Now you will have another time Tf, the flick time. A human decides the flick time. From this you derive T. The real problem here is math and 0/0 analog.
This is just bogus, Newton's first law is known to be a special case of Newton's second law. All mathematical expressions of it can be derived from second law. And "he couldn't be sloppy" is just a bad argument in general.
"We are tempted to think of the instant t=T as the first instant at which the mass moves. But that is not so. It is the _last_ instant at which the mass does _not_ move."
Fascinating stuff!
The missile knows where it is because it knows where it isn't
Pleasantly surprised to see Picard-Lindelöf on TH-cam.
A lot of people are getting hung up on the real world/physicality and that Newton's laws are not suitably at all scales. That's not the point. The point is whether even the *theoretical* Newtonian model purely as a mathematical model is deterministic as most scientists take it to be. One way to understand this is to imagine that you want to program a perfect virtual world in a special computer (which can do perfect calculus, say) with a perfect Newtonian game engine and then test if this game engine is deterministic. To truly answer this, one would need to give a precise and complete formulation of the Newtonian model first: what "manifold" we take for spacetime (R^{3,1} might be good)?, what regularity of functions are allowed for creating the virtual world?, are corners and kinks allowed for the solids in the word (the apex of Norton's dome is not differentiable)? are there elastic solids, fluids? etc. The math in the paper is well known---it is not too hard to find scenarios where unique solutions break down. This means in our perfect virtual world admitting Norton's dome, the computer running the game engine will try to solve the ODE as it is instructed to according to Newton's 2nd law and then run into a conundrum because there are multiple solutions and it must choose one---deterministic or not, a 4th law must be added (otherwise the computer will freeze/hang here). We could program it to choose one randomly---even that is actually tricky---if there are infinitely many solutions we need to assign a reasonable "measure" on the space of these solutions. For a deterministic virtual world, the 4th law needs to help pick out a unique solution---probably tricker but I haven't thought about it. The simplest way to go is to just impose sufficiently high regularity (analytic, say) on everything (disqualifying Norton's dome) so that the ODEs never have uniqueness issues.
I’m kinda confused because the answer seems obvious to me and I doubt I have the answer if indeed the issue is as big as you claim. I will read the paper when I have time just to be sure: the equation written is using a mathematical “sleight of hand” as you put it. It introduces a discontinuity in the 4th time derivative of position (the snap). This is physically impossible, because you would need an infinite variation of the jerk so, just like in other cases when using infinities, it breaks down.
Now, this is probably wrong but if anyone reads this and can tell me why I would appreciate it.
You're not the only person to comment this so I think you might be right. I wonder if any of those papers in the map she showed have this argument laid out in them.
@@Xeridanus I'm sure they're full of this argument, its rebuttal and the rebuttal of the rebuttal. It's rebuttals all the way down.
In another comment thread, a poster pointed out that Norton provided the ball bouncing off a wall as one well known example of accepted Newtonian physics where the higher order derivatives are discontinuous. And have one more example - a ball rolling off a table and suddenly experiencing gravity
@@mengmao5033 I thought about the same example, however the discontinuous derivatives come from an idealized interaction with another body moving at different speed: we are not investigating the cause of motion because we are idealizing force transfer to be instantaneous. I also thought about the table, because at face value it seems different, however, when the ball is rolling off, you are ideally removing a force acting on the ball, giving a discontinuous derivative. We can accept these discontinuous derivatives due to the fact that we understand there is a force transfer and we are simply idealizing the “start up”, but it doesn’t work without interaction between bodies at different speed. At least, that’s how I thought of it
precisely. you got it.
Really interesting video!
Small note: at 15:19 the example with the ball colliding with the barrier, it's true that you get u^2 = v^2. However, it's important to note that u and v denote SPEED, the magnitude of the velocity vector, NOT the velocity vector itself. While it's perfectly fine to have one VECTOR be the negative of another, magnitudes can only be positive (or zero), so there's no such thing as having a negative speed. The solution u = -v is in fact the erroneous one, NOT by any physical intuition but instead because of how u and v are defined.
The first solution, u = v is actually fine, keeping in mind that it only says that the initial speed is the same as the final speed-- as it turns out, this is the only thing that you can deduce from applying conservation of energy here anyway. The direction of of the ball's velocity afterwards can be found only by incorporating other laws, constraints, or assumptions.
the magnitude is -v, going into the negative direction of that line through 3d space. v * (x, y, z) + (xoff, yoff, zoff)
The ball bouncing off the wall doesn't follow Newtonian physics anyway as that would involve the wall pushing back with twice the force of the ball hitting it and not gaining any momentum from the impact. This would be creating energy from nothing.
@@piraterubberduck6056 In the limit as the wall's mass approaches infinity, the ball keeps all the energy and rebounds with the opposite momentum.
@@terenceundbud the magnitude of a vector is never negative... it is the length of a vector. At least in Euclidian space if you do not use imaginary or complex coordinates.
@@shinzon0
In a coordinate system, negative amplitudes(in scalar components) are used for vectors to choose the opposite direction in comparison of the unit vectors in the vectorial base.
You roughly say "some philosophers go as far as saying there is no one true version of Newtonian physics" at around 20:20. But isn't that particular point obvious actually? "All models are wrong, some are useful" seems to me to explain most of this conversation. I personally think the discrimination between these different models is inherently arbitrary because it's beyond humanitys epistemic reach.
Great video! Without speaking from any kind of physics background, I still think it makes sense. We create a scenario in which there is no force acting on the ball, but on the other hand nothing holding it either, not even friction, which means we have to remember that this is not a realistic scenario. Therefore intuitively it might as well roll rather than sit still. I personally find it really fascinating when someone can break our habitual thinking in such a simple way, whether it turns out to have practical applications or not🙂
Over 4000 comments in 15 hours (far more comments than seemingly most or all other videos on this channel) - this nicely reflects the mind-bending and emotion-stirring nature of Norton's Dome that I've loved about it for a long time. Congratulations on being [as far as I'm aware] the first major TH-cam channel to cover this topic!
It's not "Norton's dome", it's Maxwell's example of a Lucretian swerve. Norton plagiarized it from 1870s work by a physicist, as is typical for philosophers. The reason this is getting a lot of comments is that it is fascist-adjacent, Maxwell used this type of system to argue against full determinism in the physical laws. All these debates are obsolete since quantum mechanics.
It's not Norton's dome, it's Maxwell's example. Norton is committing PLAGIARISM, likely deliberately.
@@annaclarafenyo8185 where does Maxwell introduce this example? was it in one of his private writings or a paper?
@@annaclarafenyo8185 Could you please provide the source where Maxwell introduced this example?
@@bjornfeuerbacher5514 If Norton is supposedly a "plagiarist" (as Annaclarafenyo8185 has claimed in multiple places in comments here), then Maxwell was also a "plagiarist" by this definition. The first historical source for such an example is likely a paper by Poisson in 1806 (where it was mostly just conceived regarding the function of the force), though it didn't get a lot of attention until Boussinesq discussed it in 1879, which led to a flurry of discussion in French physics literature... and then migrated to discussion that was mostly much less formal and technical among other non-French physicists like Maxwell, who were theorizing about free will. Norton's example is really a special case of Boussinesq's "dome."
This example was so well-known in the 19th century literature (especially French physics literature) that I'd go so far as to say it was "common knowledge" (which doesn't require citation). And indeed in Norton's initial paper from 2003, he doesn't present this as anything special or novel -- merely one of many shapes that will behave in this way. It's not really anything particularly exotic physically, and there were periodic "rediscoveries" of such shapes and functions in physics literature over the years (at least two different papers in the 1990s alone that I know of).
For a summary of some of this history, you might look up the article "The Norton dome and the nineteenth century foundations of determinism" by Marij van Strien (2014). The emphasis there is on the French sources, since they discuss such things in more depth (to my knowledge) than people like Maxwell, whom I believe just used this as a vague example in some speculation on free will and determinism.
As for who should get ultimate "credit" or whether "Norton's Dome" is misnamed, see Stigler's Law of Eponymy (which was actually formulated by Robert Merton).
0:27 Nice intro, how man takes did it take to make this uncut intro?
Oh man! What the hell 😂😂😂
There's an electromagnet in the dome 🙂
1 take, 2 magnets
5:00 I might have misnderstood the setup but this is not possible in perfect newtonian dynamics: the ball will either asymptotically go to the dome or surpass it. The differential equation that governs the physics satisfies the hypotesis of the existence and uniqueness theorem.
Thank you. My comment was "is it theoretically possible to roll the ball to rest at the apex of the dome?"
I suspected it couldn't be possible, which makes this problem akin to slight of hand.
If we assume the ball CAN come to rest on the apex, then we have to assume spontaneous, uncaused motion. If it CAN'T, there is no problem.
@@gochaosgamer1759I mean if it reaches the perfect spot, it would need slight momentum to get there which would just roll over. You can approach the top but never reach. Isn’t this just limits then?
@@Bigleypit is just limits. This video is such pop sci “chocolate and wine are healthy for you!1!” type stuff. It takes a bad paper and makes it consumable for the masses to think something magical. It’s a shame she’s always been such a sham of a channel like this when she could do so much better.
@@ClementinesmWTF It is not just a limit. The ball reaches a state of rest at the apex of the dome after finite time. That's how the equations in this case play out and you can verify this if you just do some calculus yourself. The differential equation does not guarantee a unique solution because it has been constructed to avoid one.
@ so the equation has an undefined/infinite spike at the very end? Ok? Cool? All you’ve shown is that you can make a dome with an equation that is non-analytic. Like…cool. That doesn’t have implications on newtonian physics other than the equation yoy made up is not permitted-which is fine as it wasn’t something you could do in the real world or even a Newtonian world anyways.
I had asked similar question to my science teacher when I was 11. What happens at the exact time when ball hits the bat and changes its direction instantly. I was told that it does “instantly” so we don’t need to take care about it. Later, I realised that science creates an abstract model which tries to get isomorphic to real world scenario. But there is no infinitely precise abstract model that can define reality. I sometimes think of a world where even the first order formal logic fails. What would it look like?
Well for that case, and basically all real cases, the answer is simply that it doesn't change instantly. There are technically electrically forces at all distances between the ball and the bat, and when it "impacts" they reach a peak and then subside, and the forces were completely continuous and smooth the whole time.
2:47 in and this seems like an in-issue; newtons laws describe perfect theoretical environments, which ours is not; the top of the cone is not flat, we are not in a vacuum, and the ball is never perfectly at rest, it’s moving microscopically as the minute forces around it, like air, accrue in an eventually-compounding direction enough for gravity to get its hooks in. Think of Brownian motion. Am I missing something?
yess you are right
What you are missing is that the argument is that the ball will spontaneously move off the apex at some time _in this theoretical world._ There's no point talking about realworld imperfect conditions when the topic of interest is this perfect (i.e. impossible) world of a frictionless ball and dome in a vacuum only being acted on by gravity.
The point of the paper was to advance the idea that Newtonian mechanics can _also_ be indeterminate, that a Newtonian world is not fixed on rails.
Personally though I find the argument unconvincing because crux of the problem relies on a quirk of mathematics that the equation in question has multiple possible solutions. It doesn't provide a mechanism for _why_ that excitation period happens when it does, it just supposes that it does. My instinctive go-to response is that the problem is ill-posed, not that there's a hidden truth in Newtonian mechanics (which as we all know, has been rendered obsolete, but is a useful simple model for general, non-extreme cases).
I'm not entirely convinced that it is actually possible to roll the ball up the hill with just enough energy to stop at the apex. I know it feels intuitively true, but is it actually? It an idealised setup with no friction, one could argue that the ball must always arrive at the apex with some non zero velocity and therefore overshoot it. I think it's a fair question to ask at least.
@@w0tch because of that in reverse the ball would sit there forever. Just as expected.
The video raises a fascinating question about reversibility. The ball can reach the top of the dome from many different directions around the base. But the crucial question is: if the ball starts at the top, can it roll down in the exact opposite direction it came from? If we placed the ball on top of the dome, that direction is "up," which isn't reversible.
This relates to the idea of one-to-many versus one-to-one relationships. Many starting points (directions around the base of the dome) lead to a single endpoint (the top). However, when we reverse this, that single point at the top can lead to many possible directions. This "one-to-many" aspect makes perfect reversal impossible. It's like having one input (the top of the dome) leading to many possible outputs (directions of descent).
Imagine a circle around the base of the dome. Any point on that circle represents a possible starting direction. All these points converge to a single point at the top. But from that single point, the ball could roll in any direction along that same circle. This inherent asymmetry prevents a perfect reversal of the trajectory.
well all those points on a circle, where the ball starts to ascend, converge on 1 point. So there is a one-to-many correspondence as well. This is the same thing as both 2^2 and -2^2 equaling 4, which results in sqrt(4)=2;-2
It's not all that fascinating. The math in physics is there to describe what we can observe happening. We cannot observe time reversing itself, so anyone who includes that as part of their discussion is playing silly buggers right from the start.
This is an interesting approach, but for me, this is not really how time reversal symmetry works. By this logic, kicking the ball is also not a deterministic action if we observe the situation from the ending point, because the ball lying on the ground can get there from many directions, even if we describe it with a seemingly identical equation. Basically, the foot kicking the ball can be placed anywhere on a circle, facing the center of the circle where the ball will be in a "resting" position. If we reverse the action, the ball shouldn't jump back to the exact same point, it could fly to any random point on this circle.
The trick in this situation is that it is seemingly the reverse action that is establishing the situation. We claim that if we can push the ball up to the apex point, into a resting position with a perfectly executed impact, then the reversal should be a scenario when the ball seemingly starts to move from a stable resting state into a downwards rolling motion.
The reversal is only relevant to explain why we observe the situation described in the thought experiment, the direction of motion is irrelevant.
@@filipsperl I used a similar approach, considering a Mobius strip. We can tell if a strip is a Mobius strip by how many sides it has, which is determined by how many half-twists are applied. We end up with a function like s = f(t), where s is sides 1 or 2 and t is any integer. If t is odd, s = 1; otherwise, s = 2. This function cannot be reversed because it is a one-to-many relationship; it cannot be determined by setting s to any value. Your approach is more concise and easier to understand. I wanted to use an example, but my Mobius strip analogy was too complex. I like yours better.
@@BotloB I get your point. Newton's cradle is a good example of a potential one-to-one reversal symmetry, which could support this argument if we remove macro-level variables from the system (as per the thought experiment).
Really enjoyed this! I think his argument is sound, and really quite clever. Another point I think worth emphasizing... his dome example actually DOES comport with our intuition. A ball on top of a dome just "feels" unstable. If you went to the trouble of constructing such a dome and such a ball to ridiculously exact specifications, placed the ball in the exact right position with exquisite care in a vacuum chamber in the dark, I think it's reasonable to expect it might stay there for little while... but we all kind of "know" that it would, in some relatively short amount of time, roll down. The conventional explanation is that, despite your best efforts, there will inevitably be some perturbation that upsets the balance just enough, e.g., an uneven momentary change in temperature that causes an uneven expansion in the body of the ball, or a passing photon, or any number of other physical things that could change things "just enough". And, sure, that's a fine explanation, I have no problems with it. But such things are close to impossible to measure, and thus close to impossible to prove. Seems like his solution is every bit as valid an explanation, honestly.
This also breaks the Curie principle since the rotational symmetry of the dome should't allow an explanation as to why the ball rolled in any particular direction instead of any other. This is profundly problematic as Newton's laws are invariant under rotation (otherwise you would break angular momentum conservation). Big trouble in here
I thought about it too for the dome, but if we took a half-pipe sort a solution, he symmetry would be broken…
Yep, when you assume than in ideal mental scenarios things start to move by themselves, specifically contradicting the rule set of that scenario stating that should not happen, all falls apart
what a surprise
pfft
@@TheChzoronzon There is no contradiction at all in this thought experiment, you just misunderstood the problem..
@@freshrockpapa-e7799 The behavior of the ball directly contradicts Newton's first law...which is a big problem in a newtonian scenario
And the mental gymnastics the author engages in to try to avoid the contradiction are quite lame
Now repeat with me: Po-ta-to
@@TheChzoronzon Are you not only going to stalk my profile, but follow me and reply to me in every comment I make? Please get a life.
Btw you don't have to waste your genious brain in the TH-cam comments. I'm sure you know about this open problem you just learnt about (and haven't even read the actual paper) more than the entire scientific community has figured out during the last 16 years, so go publish your own instead of wasting your time here.
This to me is like answering ”how many 9’s does it take in 0.999… for it to become =1?”
There is already proof that 0.999... (infinitely many 9's) definitely equals 1. There is a Wikipedia article about it - including the proof: en.wikipedia.org/wiki/0.999... but with any finite number of nines - it's not 1.
Man I read so many comments like yours that confirm the feeling I had from the beginning of the video
All of them.
The starting point is not zero acceleration/zero gravity. The answer is that these states do not exist in the physical world. Therefore maybe zero should not exist in the equations? It always confounds me that 5x0=0 Where did the physical go, it always exists, maybe in an infinitely smaller amount.
∞
As the ball rolls up the dome, there is information about its position and momentum. After it comes to rest at the top, we know its position, but nothing about its history. Does that count as "information loss" (which we've been taught is impossible, leading to questions about information being lost in black holes)?
Just as there is not first point in time when it starts rolling there is also no last point in time where it stops rolling, the speed just is lower and lower the closer to the apex the ball is, it will never reach zero. It will never come to rest, the shape of the dome is chosen with just this attribute.
This is super cool!
It’s of particular note that the Picard-Lindelöf theorem and the Lipschitz condition were formulated more than a hundred years after Newton, when mathematical analysis was really picking up steam as people were trying to resolve other mathematical paradoxes that seemed to show up, including many debates over whether Fourier was out of it for interchanging limits and integration. The result of these debates was mathematicians developing a lot more rigor over proof methods, and arriving at much more clarity of what was underlying points of contention-for example, for Fourier’s supposed indiscretions, it turned out to be a matter of which norm on function space one was dealing with for the limits, which turns out to stem from some really deep math on the topology of metric spaces-an entire field which didn’t exist as such in Newton’s time!
So, an interpretation is: Newton’s linguistic description of his laws may have to be fiddled with-but that fiddling depends on mathematical nuances that weren’t worked out at the time! Operationalizing mechanics means choosing an interpretation in a particular mathematical structure (ie, with or without the Lipschitz condition required), and while both are valid, we now know from analysis that the resulting qualitative behavior of the physics-and hence how one would linguistically describe the physical laws-depends on that choice.
She said she researched this for weeks but we all know those weeks were really spent trying to get the ball to stay on top of the dome. 😂
what is the tool that you used to show the connections between papers at 1:42?
By the way, amazing work!!! I loved that you interviewed Mr Norton :)
These kinds of graphs appear in a few places nowadays, e.g. Obsidian has a graph view, mostly eye candy but it looks very cool!
Looks like maltego
My favorite Physics TH-cam channel! Such an engaging style, thanks for the video!
Madam....this was one of the best physics lecture I have seen i think......your way of teaching was really good...honest review.
I think the problem is assuming that the ball would rest on the dome at all. If we can subdivide time to the point where there is no instant where the ball BEGINS to move, just the last instant where it is not moving, I think it follows that we can subdivide space as finely so that there is no way in which X amount of energy can be applied to the ball causing it to roll up the dome and stop at the top. Since it cannot stop at the top, it cannot roll back down.
Certainly a ball can roll down a dome, just not from the apex. The ball (which only contacts the dome at one point) and the apex of the dome (itself only a single point) can never actually be resting on each other because they are both infinitely small. Since they can never meet due to our ability to subdivide ideal space so finely, there is essentially NO apex, and if the ball does seem to be at rest it merely a case of not being able to measure the motion because it is infinitely small. This practically required by saying "there is no moment when the ball BEGINS to move". There is no moment when it begins to move because it was already moving since it is impossible to balance an ideal ball on an ideal dome if we allow for infiite subdivision of space.
I was thinking this, but you put it into words far better than I could have.
Same conclusion that I came to.
1) some one would need to do the actual maths, but with a smooth frictionless dome, any momentum remaining as you reached the "apex" would be retained and you would roll off the far side.
2) even if it appears "perfectly" centred there is a infinitesimal level of gap between the ball and the apex so it may even remain there for years (due to the miniscule level of acceleration that close to the top), eventually it will roll. (and with a dome shape the acceleration will transition from imperceptible to "sudden" as though it started magically)
All kinetic energy (½mv²) is converted into potential energy (mgh).
@@RobertLemonOfficial Right. So if there is no time when the ball begins to move, it must always be moving, and never rested on the dome.
@@bobcostas9716 I just ran it through chatgpt and it agrees with you and says that the time equation diverges when theta (angle difference of the object and the vertical line going through the center of the dome) is 0 meaning it takes forever until the ball fully rests. Its AI so it could be wrong
14:53: Physics cares about math, but math doesn't care about physics.😃
Instantly subscribed! Your delivery is amazing, very clear.
Surely it depends on whether there is a shortest unit of time (and everything jumps to the next place instantly). If there is then this is wrong I think and the start can be measured. Or have I misunderstood? Great video by the way
quantum nature of time?
@ who knows? At some point there is either an analogue or digital underlying time thing going on, if that analogy works? Depending on which determines the dome paradox doesn’t it?
12:46 rip that kid
I did not see that coming 😅
Who did the animation around 12:30**? Really nice work
his name is Hamed Akrami :)
Which animation? I don't see one at 8:30.
@@datadude67 7:30*
@@montero.h thank you. There are a few well done animations in this video. I wasn't clear on which one was being referred to.
@@datadude67 your welcome bro
We learned about something like this in my mechanics 101 class several years before this paper. The example we had was a bucket with a hole in it. Eventually, the water drains into another bucket. But once it's done, you can no longer figure out WHEN it finished and a time reversal is no longer possible.
Yeah this relates deeply to the Arrow of Time and such... the whole premise that Newtonian processes are reversible has no basis. Not to mention that we already have so many other proofs for Newtonian Physics being incorrect about numerous things.
@seedmole The equations of Newtonian Mechanics are time reversible if you are allowed perfect precision in time and space, but not allowed infinitesimal division of time and space. But when you get anywhere close to the limits where that matters, you run into quantum mechanical limitations anyway.
but you can measure the hole, you can measure the volume of water, you can figure out how long it would take for the bucket to empty. time is a human concept. when it started, when it ended. even if you could say when those things were they are relative concepts based on when we have defined them based on the our position around the sun. For laws of physics the exact time the bucket was empty is meaningless. its not important at all and you can calculate every other property of the hypothetic scenario.
needing to know exactly what time it was when the bucket emptied would be like saying you can't know how the water felt when it was emptied out.
plus its a hypothetical. it emptied at whatever time the person made up that it would be empty.
As an observer of the macroscopic state, you wouldn't be able to infer when the water had finished draining, but if you knew the microscopic state -- the position and momentum of all the molecules involved -- then you could in fact make that inference. Run time backwards and the water in the lower bucket would sit there for some amount of time and then start leaping up and passing through the hole into the upper bucket, all without violating the rules of Newtonian mechanics.
@@hefeweizen9475 Run time backwards? Is that all there is to it?
If you rolled the ball perfectly up hill, i don't see how it can stop at the apex. The instant before the ball reaches the apex, it has velocity. At the apex, there are no forces on the ball to stop it there, so it must continue past the apex? What force acts on the ball to stop it exactly at the apex?
Very thought-provoking. The idea of time-reversing the ball, stopping it at the top, was what caught me. Infinitely many paths lead to that position, but it could also be the 2D cupula, where the top can be reached from each side. When the ball reaches zero speed at the top, the history of how it arrived there is lost, and all the driving forces (gravity and inertia) are also "erased" (balanced or zeroed). It reminds me of the information conservation principle in quantum mechanics, only in this ideal case, the information is really lost.
The information is not lost. To get a valid, complete Newtonian system, the dome is resting on a surface (as described). If you give a ball at the bottom some impulse so it comes to rest at the top, then you also imparted some force on yourself in the opposite direction (not to mention you yourself are also at some place next to the dome, revealing the direction it came from). And you then probably also imparted an impulse to that same surface the dome is on (or not and you float in space away from the dome) (presumably a planet generating the gravity?). So the whole planet ends up rotated after the ball (and you) have come to a rest (not sure about the resulting angular momentum of the planet though).
No, I think the real issue is when, the STARTING condition is that the ball is already resting on the dome and then it suddenly rolls down to some direction.
An alternative is when the STARTING condition is when 'the force applier' doesn't exist but the ball starts at the bottom of the dome but with enough velocity already to reach the top. In order to slow down, it necessarily pushes on the dome, which then pushes on the surface (planet?), which starts rotating in some direction (hence, the direction is 'remembered').
I'm not sure if that rotating surface means the ball can't sit still on the top anymore for an arbitrary length of time.
Could the surface end up with just a translational motion instead but no angular momentum?
I just realized the 'force applier (aka you)' can't exist as it would have a gravitational effect on the ball, bringing it off balance.
@@peterbonnema8913 okay, set up four domes with each perfectly scented at the corners of a square.
Then place pairs of balls in the exact centres of the left and right sides of this square where each ball in a pair can somehow impart the necessary force upon one another to have them move apart along the side of the square and up the domes such that they are centred perfectly on their respective domes.
Then you have four balls on four domes with no way to determine which direction each of the balls came from to get onto each dome.
This removes any way to definitively track the fource that pushed each ball onto the domes and it also removes any rotational forces imparted on the planet from the initial push.
I'm sure the external world can be set up in a way to equalise gravitational forces on the balls, one way could be to have the whole set up be a cube in empty space with eight balls and eight domes instead of just four, that way the gravitational forces on each ball are all pulling equally towards the centre.
Mathematically speaking the only way for the ball to stop at the exact top is if it is still under some form of acceleration. The acceleration due to gravity goes to zero at the exact top, so the only way it could stop is by friction.
You cannot roll a ball up a frictionless dome and have it stop and balance at the top. It is mathematically and physically impossible, and produces the exact contradiction in physics that this theorem represents.
Math does not rule physical phenomena, it merely describes our observations and guides our predictions. The finer our discrimination of differences becomes (the better our detection systems are), the tighter we must control initial conditions for the math (I,e, our model) to match the observations. I am troubled by time being a parameter in explaining the dome problem. As I see it, balancing a ball on top of a dome requires a point solution. Even if it were possible to achieve that precise point solution balance, the real world has perturbations outside of our ability to discriminate. Any unmeasurably small perturbation (Geologic, atmospheric, etc,) will upset the point solution and the ball will roll off the dome. My experience leads me to believe that said unmeasurable perturbations WILL occur. Predicting when such perturbations happen is the only role for any time parameters.
Here's how I see the paradox and it's connection to the known paradoxes of infinitesimal calculus:
Here's how the 'normal' time-reversed setup looks like: you have to apply the infinitely precise amount of force so that the starting kinetic energy of the ball is equal to the potential energy at the top of the dome. And apply the force precicely towards the top of the dome.
Now, there are two ways to treat that. You can just have this setup in isolation, with that precise amount of force directed precisely where you need it. Or you can recognize that the calculus of infinitesimals, and by extention the Newton's laws, are a product of a limiting procedure.
Let's think about the limits. Say the starting kinetic energy is a bit off from the precise number you need. What you see is that the ball climbs up top, has a bit of velocity still and rolls down. Now, the smaller that imprecision is, the smaller the velocity on top will be. And that ball stays on top(in a small region around the top) for longer and longer periods of time. Go to the limit, that period of time goes to infinity.
Similar argument for the direction of the force. Say the direction of the force is a bit off. The ball gets close, but not close enough. And then rolls down. The greater the precision, the closer the ball gets to the point at the top.
Taking both limits, the ball at the top is moving with a velocity that goes to zero and and passing at a distance to the top that goes to zero, staying in any arbitrary region around the top for a time that goes to infinity.
Does that create an unphysical result? Yes. But I see that as a result of taking this limiting procedure too seriously. And in physics, you don't usually treat those like that. Take thermodynamics. You work with infinitesimals, but you want those infinitesimals to still be physically large. Say you take an infinitesimal volume, it should be still big enough to contain lots of particles. Same with electrodynamics. You can work with charge densities and work with infinitesimal volumes, but you know that those infinitesimal volumes are physically large and contain lots of charges.
Because infinitesimals are a calculational tool. A tool that tells you the asymptotic behavior. How does stuff behave when you get arbitrarily close to a specific setup? And I think that's the right way to understand Newton's laws. But I thank you and thank the author for bringing this paradox to my attention. Wouldn't have thought about that this deeply otherwise
Edit: As correctly pointed out by @deinauge7894 this argument is flawed
your description would be valid for a "normal" dome like a spherical or parabolic one. But this thing is constructed as a rotated r^(3/2) curve. And the rolling time is limited by a finite number - it can not be in a small region near the top as long as you want. (Similar to a cone shape where thats also impossible)
Note that the ball also has to be infinitely small for the whole thing to work.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle.
The apex of the dome is infinitely small or a point which according to the uncertainty principle means the ball always has a non zero momentum -- the flatter the dome the less you know about its initial position.
In the real world of quantum mechanics the ball always has a momentum -- heat or uncertainty, errors.
Action is dual to reaction -- Sir Isaac newton or the duality of force.
The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual.
Symmetry breaking always causes the ball to move -- cause is dual to effect or correlation.
Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum duality.
Initial or first cause is dual to final cause -- Aristotle.
Duality via the uncertainty principle causes the ball to move.
"Always two there are" -- Yoda.
You cannot have symmetry without anti-symmetry in physics! Duality creates reality.
@@deinauge7894 thanks for your comment. Now I need to investigate this more
Yes! Calculus will not always give "correct" results. This type of acceleration is not something that is effectively encapsulated by calculus, so the results obtained merely demonstrate such a result. The claim of spontaneous motion feels similar to claiming that the area underneath a 1/x curve passing through 0 does not converge strictly because the improper integral does not convergence.
@deinauge7894 okay, I think I see what you're saying. My argument totally neglects the fact that a ball that is not at the top is accelerated so although its velocity gets arbitrarily close to zero, it doesn't stay that way.
Basically the limiting procedure I describe doesn't even approach this paradoxical behaviour at all. Which to me seems like now the question of whether to accept this paradox as part of Newtonian mechanics boils down to 'Do we accept solutions which rely on exact initial conditions and cannot be obtained by a limiting procedure?"
18:39 if the exponent becomes zero, (1/12)^0 =1 …right?
It's multiplied not to the power of.
@@simonmackenzie8571oh ok thx.
This is a storm in a teacup!
I build pendulum clocks for a hobby. They work because the period for small angles of swing is quite predictable and constant (Galileo). However, the period gets longer and longer as the angle of swing increases. In fact, the period is infinite when the pendulum bob is vertically above the pivot point. You have to look at Gauss' Elliptical Integrals of the Second Kind and use Arithmetic-Geometric Mean to get an accurate answer (Seriously, Gauss was probably the best mathematician EVER!). The pendulum has TWO equilibrium positions - the usual stopped clock position of straight down is a stable equilibrium AND unstable equilibrium position with the pendulum pointing straight up. As mentioned above, the period with the bob straight up is infinite. Suppose the angular displacement from the vertical was 1 femto-second of arc. You'd stand there looking at it for ages (eons even) before you'd notice any displacement from the vertical. OK, in the real world air currents would unbalance the pendulum or static deformation at the crystal level, etc. would unbalance it. The marble on the hill is the same situation. It is not possible to exactly place the marble so that the moments in all directions are exactly balanced. The thing sits there and obeys Newtonian Mechanics. It is just that the imbalance force is mind bogglingly small to begin with so the acceleration is also miniscule to begin with. It is not that one second it is doing nothing and then it decides to move.
Here is something worthwhile to ponder: single photon interference made simple. Thanks!
Exactly. The world IS deterministic: and not only that, but it cannot be any other way, because it is based on cause and effect. This video is just misleading clickbait.
Sorry, no, you misunderstood the point of the video.
The system you described roughly corresponds to the differential equation x''=x (with x=0 being the topmost position of the pendulum). This *does not* display the behavior discussed in the video, and is thus irrelevant. There is no solution where x=0 for t0 for t>T. If it starts in equilibrium it remains in equilibrium. This is is roughly equivalent to a dome in the shape of a simple quadratic paraboloid.
To obtain the phenomenon in question, you need to have x''=x^a where 0
@@apokalypthoapokalypsys9573
The world definitely can be nondeterministic. There is no problem with a universe with clear mathematical cause-and-effect rules, where those rules have randomness built into them. If you think of the universe as modelled by a cellular automaton, for example, no reason the evolution rules of the automaton shouldn't have randomness.
As for whether randomness actually exists, this can depend on our interpretation of quantum mechanics; but in practice, the universe definitely is nondeterministic. Decay of radioactive isotopes, for example, is completely random and unpredictable. Bell's theorem shows that the behavior seen in quantum mechanics cannot be explained with a theory of "local hidden variables", that is, an internal state which could be used to predict this behavior if known.
This whole quest seems to stem from a statement that goes unquestioned in the video. "if you accept that it's possible to nudge the ball with just the right amount of force so that is rolls up to the apex and stops there," then by time reversal symmetry, the solution that the ball must at some point roll down with no external force must also be valid.
WHY, I'd like to ask, must we accept tht it is possible to nudge the ball just so? If we are going to base our intuition on this, it is worth examining whether it is even valid to find that a ball can come to rest perfectly still atop the dome in an idealized probelm. I suggest that the space of possible "nudges" is incomplete, that there is a pinprick where the possible force vectors would hypothetically lead to the ball resting at the top of the dome. I suggest that this pinprick itself is a disconinuity of sorts, where it's simply not possible. A missing poin, where the limits converge together, but the poin itself doesn't exist. take, for example, "f(x)=x/x". The limit of f as x approaches zero is 1. but x=0 does not have a valid solution. I propose that through this sort of discontinuity, one cannot actually roll a ball such that it comes to perfectly stop at the top of the dome.
As the ball goes to the top of the dome, the velocity of the ball approaches a value very close to zero.
As the ball rolls down from atop the dome, the velocity of the ball rapidly leaves the vicinity of that same value.
The more perfectly one rolls the ball to the top of the dome, the longer the ball spends in the vicinity of the apex.
To simplify things, let's consider just the two-dimensional case. A disc is rooled up a curve so that it reaches the apex.
Instead of trying to attain the perfect position, let's frame the problem in terms of how long the disc spends at the apex.
Since we know that rolling it too hard will cause it to roll over, and too softly will cause it to roll back, let us instead concern ourselves with optimizing for a much more controllable and much less finicky variable than which way it rolls: how long does it take to leave the vicinity of the apex?
More specifically, how can we maximize the time it spends in the vicinity of the apex?
So, we can experimentally determine the rough magnitude needed to reach the point at which the direction of the disc switches. That's perhaps a good starting point. But as we tweak the exact magnitude, the amount of time spent at the apex can only *approach* infinity.
And here is where the crux of the problem lies:
We cannot solve for the case where the time spent at the apex is equal to infinity. The function we can construct is necessarily discontinuous.
If we could say that the ball could be rolled into such a state of being perfectly balanced, then yes, time symmetry would dictate that a perfectly balanced ball must roll down at some point without external forces. But I argue that we cannot in fact say that the ball can be rolled into that perfect balance.
This is the most succinct description of my problem with this model. There is simply no way to have the ball come to a complete standstill at the peak of the dome, as the peak itself is an undefined reality.
it is correct that if it's based on an assumption that's not true... then we can't guarantee the truth of the conclusion...
after all... she did call it a hand-wavy explanation...
@@ntsure2436 But at least the limit of situations that come close to the theoretical limit are not in some far removed infinity, they seem quite normal. But it would need quantum mechanics to describe that case properly - and it gives a quite reasonable answer too.
Actually time reversal symmetry does NOT imply that the ball roles back down at some future time, because in that case time would still be running in the same direction, and is not reversed. Time reversal symmetry just means the equations of motion can predict earlier states given a later state just as well as they predict later states from an earlier (initial) state. Conservation of energy says I can roll a ball so it stops at the peak of the dome. There is no law in classical mechanics that says it ever has to roll back down.
Actually with the specific dome shape in the video the time cannot approach infinity. The ball will either roll perfectly to the top after a finite amount of time (let's say 5 seconds) and then stay there forever, or it will start rolling down again after at most 5 seconds.
If the dome had a normal shape, like a half sphere, then your argument would indeed be correct.
1:20 2008 was 16 years ago stop it 😭
19:24
If it is the last instant at which the ball does not move,
the causal instinct is about the terminating cause. What made it terminate the motion?
Quantum mechanics: “hold my beer”
Interesting but I’m wondering if the treatment of time at around 17:25 is partly responsible for the weirdness as normally time would be considered a continuous variable, whereas here it’s treated as though it is discrete. As a result there appears to be a time when the ball is excited that is separate to the instant before when it is still . However if we insist upon time being continuous then the separation between these two snapshots has to be zero as we have infinite resolution on the values of time. This would then mean that these ‘two separate’ snapshots are only in fact one and so there is no moment of excitation to consider and we return to the more physical idea that the ball is either still or moving down the ramp as you’d expect.
Time isn't treated as discrete in the video. There is the last moment when the ball is still (t = T), and at every time after that the ball is rolling. So there no first moment when the ball is moving and there is no gap.
Like others have said on here, the Earth is moving, crust moves, atoms constantly are moving. Also, just the heat from your fingers on the part of the ball where you grab it can cause changes in temperature differences within the ball material itself causing minute shifts. So many factors at play really.
Many factors, in other words net external forces.
Even though this is a fun watch I would prefer a deterministic Chaos approach for this problem.
21:00 This is exactly why programming takes ages. Even when we're the one that wrote the exact rules for the computer program, even when the computer follows those rules to a T, even when we didn't want to have a complex idea, there's always more popping out of the woodwork. We don't know the full extent of the flaws or merits in our thoughts, because we only have so much internal RAM to run them.
I read through the math of the original paper. The author bent over backwards to find a solution to the equation that wasn't smooth, and didn't have derivatives with continuous functions.
This whole controversy ends when we demand that Newtonian mechanics allows only smooth and continuous solutions.
Physicists literally hate philosophers. Check out the papers on 'supertasks'. One of them concerns the spontaneous generation of movement in a tube full of balls. All that that does is to provide back-up for a perpetual-motion scam. But the difficulty does not have to be very deep. Years ago, in Amer.J.Phys, there was a calculation that 'proved' that a vertically balanced pencil would fall over. Of course it would in real life, but not in theory. The paradox was traced to a subtle point concerning the instantaneous contact-point.
Is that the case with the world? I'm not sure.
That demand prevents you from being able to solve a bunch of very reasonable problems. For example, suppose that you want to model a car driving off of a cliff. The solutions to simple models of this will be non-smooth as the forces on the car change instantaneously when the car goes over the edge.
In fact, if you only allow continuous solutions, you cannot even model two solid objects bouncing off of each other.
@@danielkane8568 Elementary textbooks are full of examples in which vehicles travel for a while at one velocity, then change to a different velocity, and so on. This leads to a sharp-angled distance versus time graph, and students are asked questions about the total distance travelled. I used instead to ask students about the condition of the passengers. They were baffled. Not one ever 'got the point'. The passengers would be dead.
What can be determined is people not reading the paper first before commenting
It may also be determined, that most research papers are behind a million pay walls, and thus not available to the public. Yet, most of the said papers, are in large part paid for, buy the very same public.
@@FrodeBergetonNilsen the papers have links above. Or didn't YOU bother to look?
I read it. And have aerospace engineering degree. I would liked to seen how he derived the solution for the arbitrary T equation as I'm rusty on diff eqns.
That being said, could it be that T is infinitely long in the 'real world' ?
what can also be determined is the inevitable [look @ me I noticed a thing in the comnents] sequence of events ;)
@@FrodeBergetonNilsen I just googled it and it is available on the first result. Also, that argument still isn't applicable. It's like leaving a review on a book based off someone's not completely accurate recap of that book just because you can't "easily" access it
The scenario seems to imply that there is a fixed time T at which the ball will roll down the dome. But is it possible to put any bound on what value T can be? And what would influence the value in multiple iterations? T does not seem like it is defined well enough to say anything about a solution involving it.
In ECONomic growth theory, similar laws of motion exist. Here (Newtonian physics), movement leads to slope, which leads to acceleration, which leads to more movement. There (Solow 1956, link below, winning the Nobel price in 1987), capital generates output, which induces savings, leading to investment and thus more capital. Here, you can rewind the time to the state of zero motion. There, you can rewind time to zero capital. Here, you have the excitement time where motion just gets going for no reason. Same there: capital out of nothing. We (Andreas Irmen and myself, Hendrik Hakenes) pointed that out in a research article "Something out of nothing? Neoclassical growth and the 'trivial' steady state" (SSRN 2006, Journal of Macroeconomics 2008). Here: controversy. Our article was cited 7 times (end of 2024), so in economics: not much of a controversy.
The story of our paper goes like this: I had studied physics and mathematics, including some differential equations (by the way: in Bonn, where Lipschitz taught from 1864 until 1903). I then moved into economics, taking a growth class with Andreas Irmen. He taught us students the Solow model, and I observed: Wait, this differential equation (the law of motion) does not satisfy Lipschitz continuity if capital is zero. So let's see what that implies... We even had a correspondence with Robert Solow: his reply to our letter hangs in my office, I can see it right now!
Links: piketty.pse.ens.fr/les/Solow1956.pdf, papers.ssrn.com/sol3/papers.cfm?abstract_id=892732
And then when everybody realizes that like it's common sense and everybody has their own crypto currency because of it, what comes next?
@petevenuti7355 There is no money or (crypto)currency in Solow's growth model. Monetary models are much more complex 😬
@heha1390 1956 I wouldn't think so.
But
Crypto is a form of capital. It spontaneously came into being. I would say both fits the serious narrative and the joke.
@@petevenuti7355 I would argue: In a growth model, capital is an ingredient for production. Machines are capital, crypto (probably) not.
@heha1390 so cash is not capital but your tools are the best capital your saying , assets that bring more value then face value.
I'm reminded of Zeno's paradoxes, in which an arrow cannot move because at any given instant it is stationary. If time is quantized, then that means there must be at some level a sharp divide between the time when the ball is moving and when it's not moving. Perhaps it is easier to consider the case when it rolls up the dome and stops exactly at the top: what does that look like, in the last analysis? One can only break motion down into pieces so small, before one must fall back on a contest, either- or, scenario. The main problem with Newton's laws is not that they are not deterministic, but that they assume a Continuum.
Calculus does tell us that an infinitely growing sum of infinitely smaller pieces get to a finite result, though. That's Newton's massive breakthrough for me.
Thank you!
I know very little of physics but for most of this I was thinking of that thing where the arrow is stationary at every moment of time even though it's very obviously in motion. Thought it was called Time's Arrow but turns out that's a whole different thing. Thanks for pinpointing my place of pedestrian ponderance
How can it matter for a ball if it's on the top of a dome? Would it start to move on a flat surface? On the top of a cone? I don't understand this video.
the specific shape ensures the equations of motion have multiple solutions
How can you calculate the exact amount of potential energy you transferred to the ball when you placed it at the apex? In my opinion, it can never be zero. Hence, it is the same potential energy that makes the ball roll down at some point. So how is there a paradox here?
The more common term for things that don't have an instant at which they start moving is 'stationary'. I'm not entirely convinced that stationary objects are moving, whether deterministically or not. If the Big T Excitation time is the bog standard 'somebody pokes it with a finger' sort of excitation, then yeah it starts moving. If Big T Excitation is some unknown indeterminate event where nothing causes it to move but it does so anyway, then it's magical what-iffery, not Physics. If Big T is "we don't know when or how, but at some T in the future it might do" then it looks awfully like a divide-by-zero scenario is approaching.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle.
The apex of the dome is infinitely small or a point which according to the uncertainty principle means the ball always has a non zero momentum -- the flatter the dome the less you know about its initial position.
In the real world of quantum mechanics the ball always has a momentum -- heat or uncertainty, errors.
Action is dual to reaction -- Sir Isaac newton or the duality of force.
The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual.
Symmetry breaking always causes the ball to move -- cause is dual to effect or correlation.
Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum duality.
Initial or first cause is dual to final cause -- Aristotle.
Duality via the uncertainty principle causes the ball to move.
"Always two there are" -- Yoda.
You cannot have symmetry without anti-symmetry in physics! Duality creates reality.
Idealism (universals) is dual to nominalism.
@@hyperduality2838 as defined at least three times in the start of this video, this video is only talking about Newtonian mechanics and explicitly _not_ including quantum mechanics.
this dome is shaped in such a way that if you roll a ball up the hill, it can completely stop on the apex in a finite amount of time. therefore, the reverse situation where the ball starts rolling *down* the hill in finite time is also allowed in the maths.
it isn't pushed down the hill, just like there is no finger to stop it as its rolling up the hill. in the uphill situation, T is the time at which the ball becomes completely stationary. in the downhill it is the time at which the ball is rolling downwards after.
The reason this is such a big deal is that the deterministicness of the **theory** of newtonian physics was assumed by a lot of people. theoretical physics as a whole seems like what you would call "magical what-iffery", so in the sense that this only has bearing on the theory, it is indeed magical what-iffery.
there aren't any divisions by zero going on, for the record.
@@chartroniumdude5870 "it can completely stop on the apex in a finite amount of time." Actually in mathematical sense -- in an ideal environment of perfectly smooth surfaces and zero friction -- the ball WILL take infinite time to reach the true apex if you give it just enough impetus to reach the true apex.
And, that is why the whole discussion is non-sense -- IOW, in the reverse scenario, you need an infinitely small force to get the ball rolling at 'zero' time.
Which also solves the 'paradox' or whatever we want to call it: No matter how subtly I move the ball (when trying to place it on the apex), I can't get it there in a finite amount of time, so the ball will always roll off (from the non-apex) in a deterministic way when I release it.
Yes, this is math, not physics
0:12 *Hypothetical world
?
Both are right
@@sese8976 No.
well it ain't called hypothetical physics
@@_pitako Maybe it should be. Either that or call things that are well established something other than "scientific theories". Perhaps then there wouldn't be people saying things like "gravity is just a theory".
As far as I understand, in physics, had no concept of equality. You can never be exactly at the top of the dome.
Any attempt to "idealize" a situation leads to mathematical problems that don't exist in reality.
This wasn't even a mathematician behind this, it was a philosopher so it is more of a philosophical problem than a mathematical or physics one.
There is no such thing as the top of the dome, because of atomic vibrations.
18:30 raising the term to the power of zero, would make it 1, not zero. Right?
Oh so this explains why i randomly fall from my bed while sleeping all the time.😂😂
Awake is dual to asleep -- different states of the same thing, consciousness.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle.
The apex of the dome is infinitely small or a point which according to the uncertainty principle means the ball always has a non zero momentum -- the flatter the dome the less you know about its initial position.
In the real world of quantum mechanics the ball always has a momentum -- heat or uncertainty, errors.
Action is dual to reaction -- Sir Isaac newton or the duality of force.
The Schrodinger representation is dual to the Heisenberg representation -- quantum mechanics is dual.
Symmetry breaking always causes the ball to move -- cause is dual to effect or correlation.
Symmetry (Bosons) is dual to anti-symmetry (Fermions) -- quantum duality.
Initial or first cause is dual to final cause -- Aristotle.
Duality via the uncertainty principle causes the ball to move.
"Always two there are" -- Yoda.
You cannot have symmetry without anti-symmetry in physics! Duality creates reality.
Idealism (universals) is dual to nominalism.
The universe always choses the solution with the sorest bum. This is canon now. 😌
@@hyperduality2838 Knew, always I did, that to the rescue, come Yoda would!
Stop sleeping on a dome :)_
Maybe, but it still doesn’t explain why you poop your pants before falling off of the bed.
Not uninteresting (the paper, not the video), but I thought this was long known, and I don't find it shocking at all. The Picard-Lindelöf theorem is usually presented alongside uniqueness violations, one of them has a ball rolling on the edge of a cylinder, perfectly balanced, but which could fall to one side at any point and still form a solution of the equations of motion. The equations of motion are merely constraints over what motions can take place, though generically, they do yield a well-determined ODE.
From classic physics in the real world, not math infinitely smooth world, the ball falls because of deformation of the material of the dome (elastic and plastic), thermal agitation/expansion/contraction and air currents.
THIS!
There is motion somewhere that causes the ball to start moving. It’s not spontaneous at all
Just because you can or measure it or see it does not mean it does not happen
Maybe over the course of the next 20 min this is going to be discussed but 2:41 in it is being proposed that there is some earth shattering concept contrary to this
As a philosophical concept like trees falling with no one there making sounds great. From the concept of teaching or learning or making a point…. This seems like the de-evolution of critical thinking in science