# Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any examples of non-integrable systems that are not chaotic?

• What is the precise definition of chaotic you are using? – ACuriousMind Aug 20 '15 at 15:58
• @ACuriousMind I was in particular thinking of deterministic chaos, where the slight change in initial conditions leads to a drastic change in evolution, but would look for guidance on this topic! :) – AngusTheMan Aug 20 '15 at 16:03
• @AngusTheMan You need to make sure what definitions are behind the terms you're using, to elaborate: when you say "integrable", do you mean the integrability of the system as defined by Liouville? In which case, if the hamiltonian system with $n$ DOF does not exhibit at least $n$ global first integrals of motion, all in involution (Poisson commuting), then the system is not Liouville integrable. For such non-integrable system, neighbouring bundles of trajectories in phase space, spread exponentially in time w.r.t. one another, which in turn means unpredictability in longterm, so it is chaotic. – Phonon Aug 20 '15 at 16:23
• @Phonon This comment really helped me, thank you. Would the exponential decay you refer to be the Lyapunov exponent? Is this a general property of the Hamiltonian formalism that it is exponential separation? (e.g. why not linear or quadratic etc)? – AngusTheMan Aug 20 '15 at 17:28
• @AngusTheMan You're welcome. The Lyapunov exponent is a possible measure of sensitivity to initial conditions, more specifically, given 2 trajectories with initial separation $\delta x(0),$ the separation at time $t$ is given by: $|\delta x(t)| \approx e^{\lambda t}|\delta x(0)|.$ The exponential divergence of paths here is characteristic of a classical chaotic system, the used formalism is irrelevant. As to why it is exponential, is another discussion, in fact in very long times it is not exponential anymore. All this is related to ergodic properties of chaotic systems. – Phonon Aug 20 '15 at 17:42

The key point here is that, any dynamical system that is not completely integrable will exhibit chaotic regimes1. In other words not all orbits will lie on an invariant torus (Liouville's torus is the topological structure of a fully integrable system), in principle a chaotic system can even have closed stable periodic orbits (typical for regular/integrable systems) for some initial conditions, the set of such conditions has measure zero (meaning the states on that orbit are only reachable from other states of the same orbit).

In order to get yourself acquainted with such concepts, I suggest looking into 2D dynamic billiards. These models are of great interest because their dynamics are solely defined by the shape of the boundary, circular, ellipsoid, stadium etc. Now an interesting example to showcase here would be the oval shaped boundary (note circular and ellipsoid billiards are regular because of their symmetries):

In the above image (by Tureci, Hakan, et al. 2002), on left you see the poincaré map2 of the 2D oval billiard (with specular reflection), and on the right you see 3 examples of different regimes of the system. This is a perfect example showcasing a system that admits only locally integrable regions. Case a) corresponds to a quasi-periodic orbit, only marginally stable. Case b) shows a stable periodic orbit surrounded by a stable island and finally case c) corresponding to the entirety of densely dotted regions of the map, is indicative of chaotic motion. For further reading, I suggest looking into some of the articles on scholarpedia, and of course not to miss this fantastic review by A. Douglas Stone.

1For example all non-linear systems that are not Liouville integrable (as explained in comments). Note that linear systems can always be solved by exponentiation. But that said one must be wary of distinctions between solvability and integrability.

2These maps are obtained by choosing a poincaré section, and finding the intersection of trajectories in phase space with this section. Such maps allow for a representation of the evolution of any dynamical system, regardless of the dynamics involved. For more intuition, see here.

• Thank you for a great answer! So the phase space of all but the simplest systems is actually a mix of different regions (integrable +non-integrable). The patch relevant being determined by the initial conditions? – AngusTheMan Aug 21 '15 at 9:45
• @AngusTheMan Crudely put, yes :) – Phonon Aug 21 '15 at 12:53
• For fun, here's a clip of a pool variant with an elliptically shaped table, where aiming and potting is still perfectly possible, thanks to its integrability ;) – Phonon Aug 22 '15 at 4:59