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EDIT : I formerly claimed something incorrect in my question. The Liouville measure needs NOT be ergodic on hypersurfaces of constant energy. Also, I found out that NO hamiltonian system can be globally ergodic.

So the new formulation of my question is now this :

Do we call chaotic any hamiltonian system that exhibits the usual chaotic properties on each hypersurface of constant energy (e.g. ergodicity, mixing, positive entropy, positive Lyapunov exponent, etc.) or do we require a "complicated" geometry of those hypersurfaces ?

For example, imagine a system whose hypersurfaces of constant energy are very simple, like planes $z$=constant, but with complicated, chaotic behaviour on each of those planes. Would you call that system chaotic ? Thank you for your thoughts !

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2 Answers 2

up vote 4 down vote accepted

I think you are referring to Kolmogorov-Arnold-Moser (KAM) theorem. This theorem permits to have a clear definition of what a well-behaving Hamiltonian is and so, also how to define Hamiltonian chaos.

One can make things quite easy to explain by considering an integrable system undergoing the effect of a small perturbation. An integrable system will be described by a set of invariant tori. This means that a set of action-angle variables exists $(I_i,\theta_i)$ such that your system will be described by the Hamiltonian $H=\sum_i I_i\omega_i$ and your equations of motion are simply $\dot I_i=0,\ \dot\theta_i=\omega_i$. When you perturb this system, with a perturbation $\epsilon V({\bar I},{\bar\theta})$, with $\epsilon\rightarrow 0$, KAM theorem states that your tori do not change too much and, for almost all your initial conditions, the motion will happen on slightly deformed tori.

The point is that "almost all". Due to resonance conditions $\sum_im_i\omega_i=0$, you cannot grant everywhere that the motion is always bounded. There will be small volumes, small because the perturbation is small, for which the motion is not bounded and they will be covered uniformly satisfying the condition for ergodicity to hold, except for the value of the energy. These regions where the motion is not bounded are those in which chaotic motion happens. As already said, when $\epsilon\rightarrow 0$, they form a set of null measure. In this situation, the initial tori of the integrable system are still there.

When you start to increase $\epsilon$ you will see an interesting phenomenon: These regions of chaotic motion become more and more larger till the point to cover all the available phase space. All the initial tori are destroyed. In this situation the system becomes fully chaotic and the phase space is uniformly covered by the system, being motion completely unbounded, if we exclude the fact that the energy must have a definite value.

But we can increase $\epsilon$ to the point to cross toward the opposite limit $\epsilon\rightarrow\infty$. In this case, the limit of a very strong perturbation, a quite interesting phenomenon can happen: Tori reform! This could not be generic but, for the most known systems, the limit of a very strong perturbation describes again an integrable system. Chaotic motion disappears and tori reappears.

You can see an evidence of this here. You can change the intensity of the perturbation and the system will transit from regular to chaotic and back to regular motion. The system I consider here is that of a harmonic oscillator moving in a plane wave. This kind of system is interesting in studies of plasma physics. The reference is this, appeared in the Journal of Mathematical Physics.

The conclusion to be drawn from this is that a system can be fully ergodic just in a small window of the parameter space. If you want to understand the ideas underlying an ergodic behavior in a large system you have to turn your attention to quantum mechanics and to the Lieb and Simon theorem.

For a more historical overview, I have found this paper. The reason to cite it is to remember two pioneers in these lines of research.

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So you say that regions og the phase space for which the tori disappear are chaotic. OK. but what about systems that are not perturbations ? I'm thinking for example of the double pendulum (unsure about the correct term, sorry), which is said to be "chaotic". –  Glougloubarbaki Dec 8 '11 at 17:05
If by double pendulum you mean this: en.wikipedia.org/wiki/Double_pendulum, this is a perfectly legal KAM system. You can check it e.g. in ias.ac.in/pramana/v73/p453/p453.pdf with relative comments. But if the system is integrable you will get no chaotic behavior otherwise chaos is always a behavior that sets by increasing some parameters beyond a threshold. This is true also for Hamiltonian chaos as shown by KAM. The interesting news is that this chaotic behavior could disappear and ergodicity is granted just in a limited range of parameters. –  Jon Dec 8 '11 at 19:01
right, but then what is the parameter for a double pendulum ? the length of the second pendulum ? –  Glougloubarbaki Dec 8 '11 at 20:37
You are almost right. The parameter is $\gamma=m_1gl_1/E$, being $E$ the energy that is fixed through the initial conditions. Chaos sets in at $\gamma\approx 0.1$. Check the paper I cited to you by Bender et al. and refs. therein. –  Jon Dec 8 '11 at 21:57
  1. Yep: Alfredo M. Ozorio de Almeida wrote about this: http://books.google.co.uk/books?id=nNeNSEJUEHUC&pg=PA60&lpg=PA60&dq=hamiltonian+chaos+liouville+measure&source=bl&ots=63Wnmn-xvT&sig=Z0eRtIQxmdQvgWUcLBab7ZJ9y-U&hl=en&ei=0EXfTuvzJcOG8gP5mZjaBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDcQ6AEwAw#v=onepage&q=hamiltonian%20chaos%20liouville%20measure&f=false

  2. What is meant by Chaos: Laplace said, standing on Newton's shoulders, "Tell me the force and where we are, and I will predict the future!" An elusive claim, which assumes the absence of deterministic chaos: Deterministic time evolution does not guarantee predictability, which is particularly relevant for mechanical systems whose equations are non-integrable - common in systems which have nonlinear differential equations with three of more variables. Knowing this, we enter the realm of physics: In the hamiltonian formulation of classical dynamics, a system is described by a pair of first-order ordinary equations for each degree of freedom, so in addition we re-impose the conditions from deterministic time evolution (the nonlinear differential equations) and given the space is constrained: Ergodicity!

Hamiltonian Chaos: http://www.phys.uri.edu/~gerhard/hamchaos.html

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I don't really understant your point for 2. Are you saying that any ergodic hamiltonian system is chaotic ? –  Glougloubarbaki Dec 8 '11 at 17:01
Also, I found a partial answer to my question in your reference, but it kinda contradicts what you said : no hamiltonian system can be globally ergodic ! –  Glougloubarbaki Dec 8 '11 at 17:14

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