# Classical Chaos: Can a Particle Move Chaotically and Be Subject to Force?

• I
In summary: The connection between chaos and phase space trajectories is that as a system evolves over time, its phase space will change, as will its trajectories. This is an important concept in chaos theory, as it provides a way of understanding how a system can be sensitive to initial conditions.
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Suppose we are talking about a purely classical phenomena (OK, nothing is purely classical, but suppose we consider quantum effects as insignificant, that is, we ignore them). In this context, I came across someone talking about "a particle in chaotic continuous motion as the particle is subjected to a force". For me, this sounds impossible to take literally, since chaotic in its strict sense (not in the sense of hidden variables) would mean the lack of all predictability of its motion beyond the light-speed barrier (that is, its motion would be random, but not necessarily all factors would be random, as in "random variable", except perhaps the ) , and the force would add the predictability to the motion. So, the questions:
is it possible to talk of a classical particle being both literally chaotically moving and subject to a force at the same time? If so, does this mean that "chaotically" and "randomly" are different concepts? If not, then what better term would there be for a particle whose motion would be chaotic/random were it not for the force, but then under the influence of the force, some aspects of its motion would be predictable and others not? (For example, in a fixed three-dimensional space, the path traced out in a certain plane was determinate, but the motion in the third dimension was literally chaotic?) "Determinate with caveats" doesn't work, and "restricted chaos" sounds like an oxymoron.

chaotic in its strict sense (not in the sense of hidden variables) would mean the lack of all predictability of its motion beyond the light-speed barrier
I am not sure what the part I highlighted in red is supposed to mean to the reader of your prose.
Can you explain, or perhaps re-phrase.

Chaotic does not mean lack of all predictability.
It means how far into the future can we predict how a system will perform, or what state it will be in, with a definite certainty.
The weather is an example of a chaotic system, wherein we can say with high certainty that in 5 minutes from now the conditions will be the same. As the time frame extends, the certainty drops.

Random means that we cannot predict the outcome of a system.

Why are you searching for new terms? They seem to have been accepted in the scientific community as having a known meaning, even if in everyday speech people will use the two as being one and the same.

Thanks for the clarification, 256 bits, and the link. That answers my question. (I go by the old "I'm told I should learn by my mistakes. Since I want to learn, I guess I'll have to go out and make some mistakes.")

256bits
is it possible to talk of a classical particle being both literally chaotically moving and subject to a force at the same time?
Yes. Chaotic motion is a classical concept. It does not mean random, it means hard to predict. It means that small errors lead to large deviations in the future.

A good example of a chaotic system is a water mill with slowly emptying buckets. If the fill rate is slow, then this water mill will be non chaotic. It will turn at a steady rate, and if you come back an hour later you can accurately predict its state. If the full rate is fast then it will be chaotic, and if you come back an hour later your prediction of its state will be very inaccurate.

Thanks, Dale. The difference between random and chaotic was one which was not clear to me when I wrote the post. But if I now understand correctly, when your wrote
Dale said:
Yes. Chaotic motion is a classical concept. It does not mean random, it means hard to predict. It means that small errors lead to large deviations in the future.
both conditions (unpredictability and high sensitivity to initial conditions) make up the definition of chaotic, whereas only unpredictability is sufficient for the definition of random. I would guess that one could formulate the following: the chaotic system is one in which small changes in initial conditions create a tendency towards randomness in the state change. Would this be a fair guess?

the chaotic system is one in which small changes in initial conditions create a tendency towards randomness in the state change. Would this be a fair guess?
Yes. Are you familiar with the concept of phase space and phase space trajectories?

Dale said:
Are you familiar with the concept of phase space and phase space trajectories?
Only superficially. My rough definitions are: (a) a phase space for a system is a graph whose axes are all the variables (momentum, position, etc.) needed to describe the system; a phase space trajectory is the graph of time versus the rest of the variables. My guess of the connection with chaos is that the phase space trajectory would appear chaotic, and if stable, a fractal, for example a Lorentz attractor. However, my knowledge doesn't go deeper than this, but I would be grateful for any better indications.

Your rough definition is fine. So for a pendulum, one axis would be the angle, and the other would be the angular velocity. Any possible state of the pendulum is represented by a point in that phase space. And the laws of physics tell you how the system moves from one state to another, tracing out a line in the phase space.

So, instead of thinking about starting with a single point in phase space, think about starting with a small area of points, each one associated with its own trajectory line. One characteristic of phase space is that the trajectory lines never cross, split, or join. This leads to the fact that if you take the area of your initial patch, let them evolve for some time, then the area of the final patch will be the same as for the initial patch and it will still be all connected.

Although the area may be the same and it will be connected, in a chaotic system it can become very long and skinny and stretched out. This is why it is difficult to predict the behavior of a chaotic system far into the future. Trajectories that start out arbitrarily close can wind up arbitrarily far away in phase space.

Thanks very much, Dale. If I understand correctly, you are pointing me in the direction of an application of Liouville's Theorem. That is very instructive; I shall take that as my starting point for looking into this problem more deeply.

Yes, that is correct. Let us know if you have any other questions on the topic.

There are a lot of different definitions of chaos in finite dimensional dynamical systems. And there are a lot of different effects those are treated as chaos. For example the double pendulum in standard gravity field does not have another analytic first integral that would be independent on the energy integral almost everywhere. This effect is considered as chaos.

Old posts come back to haunt me; having re-read the thread before reading wrobel's new entry (for which thanks), I noticed that I never answered a question posed in #2 above

256bits said:
"chaotic in its strict sense (not in the sense of hidden variables) would mean the lack of all predictability of its motion beyond the light-speed barrier "
I am not sure what the part [beyond the light-speed barrier] is supposed to mean to the reader of your prose.
Can you explain, or perhaps re-phrase.
Yes, I see that was poorly phrased. I meant to say that the classical motion cannot be entirely unpredictable, in that we can at least put an upper limit on its speed. "beyond" in the sense of "except for".

Now I have a question
wrobel said:
that would be independent on the energy integral
Was that supposed to be "independent of the energy integral", or "dependent on the energy integral"?

256bits
Was that supposed to be "independent of the energy integral", or "dependent on the energy integral"?
that means: any analytical first integral is dependent with the energy integral almost everywhere

Way back in the 70's or 80's, when Chaos became a trendy thing to be involved in, everyone was looking at the Mandelbot Set, which is a graphical representation of the result of a very simple Mathematical Operation. You don't need to understand the Maths of it or how the programs are written but it gives a very pretty display of Chaos at work. Just enjoy what the link shows you. No need to understand the Maths but you can get the message about Chaos. It's not random (the values are totally determined by the algorithm - the patterns all share characteristics all over the area and there are variations between.
The image in this link shows the values at pixel level that are produced by the Mandelbrot algorithm (?) for a given step size in the input values. If you select a small part of that image (preferably on the transition with the black area), you get a zoomed-in part of the pattern with smaller step sizes. The higher the zoom level, the finer the detail and the chaotic nature of the algorithm takes you further and further down into the process as the step sizes decrease. There are other controls, such as number of iterations which produce variations.
The similarities between some of the patterns that you can see over the image shows variations on basic patterns. Weather forecasting computers can recognise when the weather is varying in a chaotic way and they then 'know' that the forecasts can be very unreliable - other times the non-chaotic behaviour gives confidence in the predictions.

sophiecentaur said:
No need to understand the Maths but you can get the message about Chaos.
Unfortunately it is a delusion. If you do not understand math you can get message but you can not read this message. If you do not understand math you can not even understand the relation between the Mandelbrot set and dynamics. Further, chaotic behaviour is far not reduced to the fractal effects only

Last edited:
wrobel said:
Unfortunately it is a delusion. If you do not understand math you can get message but you can not read this message. If you do not understand math you can not even understand the relation between the Mandelbrot set and dynamics. Further, chaotic behaviour is far not reduced to the fractal effects only
I agree that it’s a lot harder than just looking at a picture (of course). But one of the lessons about chaos is the existence of such functions. They don’t have to relate to mechanics to have a meaning.
Those patterns can be appreciated on their own.
The simpler ‘foxes and rabbits’ graphs can also be viewed without a practical context. Even just the message that Chaotic and Random have different meanings is worth while getting across, imo, and the Mandelbrot graphics are just fun. The deeper understanding is likely to escape many people for ever - just as with Electricity, Quantum Mechanics and Fluid Dynamics.

## 1. What is classical chaos?

Classical chaos is a phenomenon in which a system, such as a particle in motion, exhibits unpredictable and seemingly random behavior, despite being governed by deterministic equations.

## 2. Can a particle move chaotically?

Yes, a particle can move chaotically if it is subject to chaotic forces or if it is in a chaotic environment.

## 3. Can a particle be subject to force while moving chaotically?

Yes, a particle can still experience the effects of force while moving chaotically. The chaotic motion may make it difficult to predict the exact trajectory of the particle, but it will still be influenced by forces acting upon it.

## 4. What factors contribute to classical chaos?

Classical chaos can arise from a combination of nonlinearity, sensitivity to initial conditions, and complexity in a system. These factors can lead to chaotic behavior even in simple systems.

## 5. How is classical chaos different from quantum chaos?

Classical chaos refers to chaotic behavior in systems described by classical mechanics, while quantum chaos refers to chaotic behavior in quantum mechanical systems. While both involve unpredictability and complex dynamics, the underlying principles and equations governing the two types of chaos are different.

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