Was Chaos really Newton's death ?

[mnedelko]

Dear Physics Forum community.

Please allow me tro preface the following by stating that I am a novice to the concept of chaos theory. Although I do understand the general concept behind Turing, Bellousov, Laurenz and Madelbrot's concepts, I am not entirely familiar with the mathematical basis of their scientific discoveries.

However, reading about the implications of the Chaos Theory I feel I am missing one important point and I hoped you would be able to shed some light on my question.

The Chaos Theory was heralded as the end of the Newtonian dream. To my understanding, the interaction between the concepts of feedback and self-similarity disallow for a predictable outcome to occur for any given process driven by a nonlinear mathematical equations. As such, science diverged from the idea that we would be able to reliably predict the result of chemical, physical and biological equations with nothing random in them, becuase we would never be able to know the starting point accurately enough.

But why? Why wouldn't we be able to know that starting point?

Letz assume for a second that we would know everything about the start of the universe: The exact location speed and direction of every single particle at that stage... wouldn't that mean that we would be able to predict the future of everything that will ever happen in the universe? Wouldn't that enable us to predict where and when feedback and self-similarity and even evolutionary processes would occur at any given time (assuming that we had a super-computer to process all that data). Doesn't the universe pehave as a giant simpulation with predictable outcomes (at least in principle)?

What would throw Newton's dream into disarray? Or is chaos just a science's way of approaching the vast complexity occurring in our universe with a less ambitious/ realistic approach?

I would love to get your feedback on this (however, please bear in mind that I am not an expert).

 

[Filip Larsen]

Welcome to PF!

The dynamics you described is often called deterministic chaos to emphasize that the underlying dynamics in theory is deterministic. If you have two absolutely identical and isolated systems you are correct that they would both in classical mechanics (i.e. ignoring quantum effects) exhibit the exact same state trajectory. In other words, the underlying dynamic equations do not themselves introduce any random elements.

However, as soon as we start talking "real physical mechanics" instead of just "classical mechanics", there are a number of issues that "weakens" the deterministic part of a chaotic system:

It is not practical possible to know the exact state (infinite precision) of a system. If you just continue to improve your measurement precision you will end up banging against the quantum mechanical uncertainty principle [1] which effectively limits the precision you can measure the dynamical state of a system. For instance, for a system of particles you need both position and velocity of all particles to determine the future trajectories of the particles, but the uncertainty principle "ensures" that you cannot know both at the same time with arbitrary precision.

And even if you (by magic or assuming a classical, non-quantum, world) knew the exact state of a system then you would in general not be able to calculate the exact state trajectory of the system since you in practice only can calculate (i.e. predict) the trajectory using approximative methods like numerical calculations and such methods introduces small errors that, however small they may originally be, in chaotic system over time can grow exponentially until the error match the size of the system itself.

So, I'd say its a combination of both our inability to know the exact state of system and our inability to calculate exact trajectories that puts an effective limit to the "infinite determinism" found in linear classical mechanics.

Note however, that chaotic dynamics does not mean we cannot predict anything. It just means we cannot predict exact trajectories. We are still able to predict quite a lot about the general characteristics of a chaotic system and there is no theoretical limit (apart from physical limits like the uncertainty principle) as to how accurate we can calculate how probable a particular trajectory is (like when predicting that it with 43% probability will rain more than 2 mm of rain a particular day at some particular place on Earth).

Combining the stochastically nature of quantum mechanics and chaotic systems, I'd say that probabilities is something you have to get used to are inherent in nature :)[1] http://en.wikipedia.org/wiki/Uncertainty_principle

 

[AlephZero]

To my understanding, the interaction between the concepts of feedback and self-similarity disallow for a predictable outcome to occur for any given process driven by a nonlinear mathematical equations.

That is not true. There are many processes described by nonlinear equations which are just as predictable as if they were linear.

Letz assume for a second that we would know everything about the start of the universe: The exact location speed and direction of every single particle at that stage... wouldn't that mean that we would be able to predict the future of everything that will ever happen in the universe? Wouldn't that enable us to predict where and when feedback and self-similarity and even evolutionary processes would occur at any given time (assuming that we had a super-computer to process all that data). Doesn't the universe pehave as a giant simpulation with predictable outcomes (at least in principle)?

Even if you simplify the situation by ignoring quantum mechanics (i.e. you are no longer talking about the universe as we observe it), one basic problem with this idea is how you could represent the "exact" values of every particle in the universe, inside the actual universe. Think about it - on average, the amount of "stuff" that is available to store the "exact" position and velocity of each elementary particle is ... one elementary particle. Catch 22 applies.

Or is chaos just a science's way of approaching the vast complexity occurring in our universe with a less ambitious/ realistic approach?

I would replace "less" and "more" in that sentence. Newton's philosophical position could never be implemented in practice, even if chaotic systems did not exist. On the other hand, studying chaotic systems has led to understanding better what we CAN compute, and how to use that information to do something useful. That is already being done practical "down to earth" engineering and technology situations.

 

[Studiot]

The dynamics you described is often called deterministic chaos to emphasize that the underlying dynamics in theory is deterministic. If you have two absolutely identical and isolated systems you are correct that they would both in classical mechanics (i.e. ignoring quantum effects) exhibit the exact same state trajectory. In other words, the underlying dynamic equations do not themselves introduce any random elements.

I can't agree with this statement.

Chaos is inherent in many of the equations of mechanics, from the two equilibrium position of a pendulum in motion to the instability of a tumbling rectangular object to give some simple examples.

What is true to say is that this was not fully realized in the time of Newton.

Further I don't think it was Newton who believed in the 'clockwork universe' concept - he was to mystical for that.

 

[Filip Larsen]

I can't agree with this statement.

You cited three statements of mine. Perhaps you can tell exactly which one you don't agree with and why?

 

[epenguin]

A dripping tap is under certain conditions, and according to sound or plausible physical laws and models chaotic dynamics - it also sounds like it and drives you mad.

But under other conditions it is so regular it has been used as a clock, though can still be rather maddening.

Not everything then is for practical purposes chaotic or unpredictable.

I suggest chaos is all very interesting, but there would hardly be any science, nor engineering, if everything were chaotic. If causes and effects were not often in a sense proportionate.

Then science does involve prediction. I am not sure of this, but without prediction do we not have no falsification? Also if we correctly predicted an effect beforehand we do regard that as more satisfactory and convincing than if we have rationalised it after the event (though I believe philosophers have debated the justification of this feeling. Offhand I'd say that the success of post hoc rationalisation is built into the procedure so being right is cheating whereas our predictions are more unlikely to be right by accident - hence also the importance of quantitative prediction which gives less % chance of being right by accident.)

But who said that all science required prediction, that everything had to be predictable, that that is a necessary component of science or of all science? That is not implied by the methodological advantages of prediction I mentioned above. We seek insight, understanding, of the nature of nature, prediction was a tool in this and our reward enabling applications. At length we got so much insight that we can predict unpredictability.

(BTW the dream of complete prediction is more usually expressed by a quote of Laplace not Newton. Didn't Newton even wonder whether it would really be true in the end even of the model solar system? Whether Laplace seriously believed in it, or was just expressing a kind of ideal that classical dynamics seemed to suggest - typical first slightly philosophical page of a textbook never thereafter referred to - I don't know.)

 

[Studiot]

You cited three statements of mine. Perhaps you can tell exactly which one you don't agree with and why?

All three suggest that for any given (soluble) problem in dynamics classical equations can be written that unequivocally describe what happens if a system is started in some given state.

So for instance in my examples

1) The pendulum has two positions of equilibrium viz at the bottom of the swing and 'inverted'. What equations of dynamics unequivocally determine which will occur, or more usually why the inverted position is not observed?

2) Take a brick shaped object with three different lengths of side, say a>b>c.
Write down the dynamical equations that determine the revolutions of the brick if spun about and axis through faces with sides bc and released.

 

[Filip Larsen]

All three suggest that for any given (soluble) problem in dynamics classical equations can be written that unequivocally describe what happens if a system is started in some given state.

Yes, that is one of the tenets of classical mechanics in which all laws and equations of motion on a fundamental physical level are deterministic (as opposed to stochastic) in nature, thus enabling predictability.

So for instance in my examples

1) The pendulum has two positions of equilibrium viz at the bottom of the swing and 'inverted'. What equations of dynamics unequivocally determine which will occur, or more usually why the inverted position is not observed?

2) Take a brick shaped object with three different lengths of side, say a>b>c.
Write down the dynamical equations that determine the revolutions of the brick if spun about and axis through faces with sides bc and released.

Based on these examples I must say I fail to see what you disagree with.

Both examples have completely deterministic dynamic equations of motion in the context of classical mechanics. The systems are also non-chaotic, so if I knew the exact state of the systems I could, with access to sufficient precision, calculate their trajectory for all time.

For example, tossing a brick in the air and onto a surface is not a chaotic system, since it has exactly six stable point-like attractors in phase space. I agree that the transient trajectory for the system exhibit sensitivity on the initial conditions, but that is not really chaotic motion since knowing the exact initial state still allow anyone to calculate the trajectory for all time using only a "finite" amount of precision.

I still maintain my claim that in the context of classical mechanics, systems are a) governed by completely deterministic equations of motion and b) have predictable trajectories if the system is non-chaotic. Do you disagree with either a) or b)?

 

[Studiot]

Yes, that is one of the tenets of classical mechanics in which all laws and equations of motion on a fundamental physical level are deterministic (as opposed to stochastic) in nature, thus enabling predictability.

I'm sorry, which of Newton's (or other) law states that?

I sugggest you try doing exactly what I prescribed (though not with a brick) with the spinning object.
You might be suprised at the result. I certainly was when the chaotic phenomenon was first introduced to me.

I should perhaps add that the chaotic instability is derivable directly from Euler's three equations, which on the surface appear identical in form and deterministic.

 

[Filip Larsen]

I'm sorry, which of Newton's (or other) law states that?

They all do by being formulated in a deterministic mathematical framework, meaning that a given system will always result in the same trajectory given the same initial state. This is true even for chaotic systems.

Perhaps "deterministic" means something else to you? A fairly concise definition of determinism as I have used it, is given by [1] and [2].

I sugggest you try doing exactly what I prescribed (though not with a brick) with the spinning object.
You might be suprised at the result. I certainly was when the chaotic phenomenon was first introduced to me.

While you no doubt easily could find systems that I have never tried to actually model and "solve", your examples are not among them. I may add that I did once upon a time major from a physics department doing research into chaos and non-linear systems, so I have actually seen and researched a bit of such systems.

But to make a long story short, I suspect you equate presence of sensitivity on initial state with presence of chaos. If you care you could perhaps find time to read Weisstein's short "synopsis" of chaos [3] and tell me if you disagree with that. If you agree with him then we must at least agree at some level, and if you don't agree with him, ... well, then you may as well take it up with Weisstein :tongue: If you agree with Weisstein, but still think I'm talking rubbish, then I must be really bad at expressing myself or you must be applying some untraditional interpretation to the terms I use.


[1] http://en.wikipedia.org/wiki/Deterministic_system
[2] http://mathworld.wolfram.com/Stochastic.html
[3] http://mathworld.wolfram.com/Chaos.html

 

[Studiot]

I don't like discussing with non present personae, who can't answer for themselves.

I suggested that it is impossible to predict the rotational state of the 'brick' given the known starting conditions and have posted the maths to prove it before now.

Since you think this prediction can be made I asked for the maths to make it, a request I repeat.

The members in this forum (including at least one other physicist) were more willing to delve deeper. Post#14 has the mathematics.

http://forum.allaboutcircuits.com/showthread.php?t=33367&highlight=euler

 

[cosmik debris]

I too studied chaotic systems in the early 90s and what Filip has described is my understanding also.

 

[Filip Larsen]

I don't like discussing with non present personae, who can't answer for themselves.

If you a referring to my wise-crack about taking it up with Weisstein then I surely hope you noticed the smiley. If you're not referring to my wise-crack, then I don't know what you mean by "non present personae".

I suggested that it is impossible to predict the rotational state of the 'brick' given the known starting conditions and have posted the maths to prove it before now.

I guess you are talking about a torque free absolute rigid spinning body then. My initial comment with the 6 stable point-like attractors was made with the assumption (unstated on my part) that you meant a real box in a gravity field with energy dissipation so that it sooner or later would end up at rest at a flat surface with one of its six surfaces pointing up (similar to a die toss), and that you meant predicting which side was impossible in this model. Since you apparently aren't referring this model then please disregard my comment on the 6 stable attractors.

So, if we in the context of classical mechanics talk about a torque free and absolute rigid (i.e. no energy dissipation) spinning object with three unequal principal moments of inertia, I agree that there exists a peculiar, but degenerate, case where spinning exactly about the middle principal axis will seemingly make it impossible to determine which way the object will tumble since the situation is perfectly symmetric. (This is similar to dropping an infinitely thin needle exactly over an infinitely thin vertical sheet of foil and then be unable to predict from the model to which side of the foil the needle will fall since the situation is perfectly symmetric and "choosing" a side require break of symmetry.) If it is this case you are thinking about when you say it is impossible to predict a tumbling brick, I will like to emphasize that this is a degenerate case that not in itself classify a system as being chaotic as you seemed to claimed in post #4 (you recently mention this as "only" unpredictable, so I'm not sure if you still claim tumbling to be chaotic).

If this is not the situation you were thinking about, then I don't know what you mean (I'm not able to see the math you reference to as it requires me to register with that forum before allowing me to view the pictures containing what I suppose is the math).

Since you think this prediction can be made I asked for the maths to make it, a request I repeat.

Perhaps you are familiar with Euler's law expressed using quaternions? Doing that you can express the rotational state of a rigid body in a special 4-state vector (a quaternion) that when applied to Euler's law give rise to 4 first-order linear ordinary differential equations and such a linear system cannot exhibit chaotic motion. And since the translation between quaternions and, say, the well-know Euler angles only involves straight-forward trigonometry there is no way a non-chaotic trajectory in quaternion space can translate into a chaotic trajectory in "Euler angle space".

The members in this forum (including at least one other physicist) were more willing to delve deeper. Post#14 has the mathematics.

http://forum.allaboutcircuits.com/showthread.php?t=33367&highlight=euler

Like mentioned above, I am not able to view the math you refer to. If you think it will make a difference please feel free to post the math here. Or perhaps we should in that case continue in a new thread with a link in this thread as I'm getting a feeling that we no longer are on the topic of this thread.

 

[Studiot]

Like mentioned above, I am not able to view the math you refer to

Sorry I didn't realize the attachments wouldn't come out.
I'll work on it.

I also realize I referred to the wrong faces in my original statement. It is the axis on the intermediate size faces that is unstable ie faces ab.

 

[Studiot]

Found I still had the attachments on my pc and here at PF

https://www.physicsforums.com/showthread.php?t=454595

 

[Filip Larsen]

Found I still had the attachments on my pc and here at PF

https://www.physicsforums.com/showthread.php?t=454595

Ok, thanks, I can see the attachment in that post.

What you have proved there is that rotation about the middle principal axis is unstable for a rigid body, a characteristic that is described in many textbooks on classical mechanics, see for instance Goldstein [1] who explains it using Poinsot's geometric description of the rotation.

Note, that while having an unstable fix-point (or manifold) in phase space is a necessary condition for chaos, it is also not a sufficient condition (see for instance [2] for a nice description of stable and unstable manifolds). You could say (with a bit of hand waving) that an unstable manifold is not a sufficient condition for unpredictability in the sense that the "exponential growth" you mention in your proof is only exponential in a linearized region around the unstable point, so globally there is no trajectory that "forever" go through an exponentially growing part of the flow. A trajectory that passes close by the unstable fix-point will "experience" alternate exponentially compression followed by exponential expansion each time it passes by, which on a global scale cancel each other out on average.

Or saying it in yet another way, given an initial state it is possible to calculate the trajectory for all time for that spinning body to any given accuracy using "only" finite precision, which in my book makes the motion predictable.

I think I will rest my case.[1] "Classical Mechanics", Goldstein, Addison-Wesley, 1980.
[2] "Nonlinear Dynamics and Chaos", Thompson and Steward, Wiley, 1986.

 

[Studiot]

which on a global scale cancel each other out

on average.[/

QUOTE]

The whole point of determinism in my book is that it does not encompass 'on average'.
To me determinism must perforce yield identical results every time.

In the long run we are all (life) are dead. But that doesn't make the emergence and disappearance of life deterministic.

 

[Filip Larsen]

The whole point of determinism in my book is that it does not encompass 'on average'.

I never said anything about determinism only working "on average", please read my post more carefully. I mentioned that you cannot use the presence of exponential divergence around a small linearized region to conclude that any trajectory through this region will be "globally" exponentially diverging as well, and specifically so for this case of a spinning rigid body since the "exponential growth" you have proved is canceled or countered by an equal amount of "exponential compression" when the trajectory head into the region around the unstable fix-point on its next passing by.

I have attached a picture of Poinsot's geometric model for the rotation of a rigid body, and there you can easily see than all trajectories are simple curves, even those near the middle principle axis. Globally there is no "exponential growth" so if you start two trajectories arbitrarily near each other the separation between the two trajectories will stay bounded.

To me determinism must perforce yield identical results every time.

Indeed, and it does so by definition. I'm glad we agree.