Why are we sure that integrals of motion don't exist in a chaotic system?

Question

The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$.

Why are we sure that no other, independent on $E$, integrals of motion exist in this system? One can assume existence of some, perhaps infinitely complicated, function $I(x,y,p_x,p_y)$, which is conserved and independent on $p_x^2+p_y^2$. Why is this assumption wrong?

In other words: the simplest prototypical examples of integrable billiards (rectangular, circular, elliptical) have some obvious symmetries allowing us to find two independent integrals of motion. What if, in some other billiard, such integrals do also exist, but are not so obvious and have no simple analytical form? How can we distinguish two situations:

1. there exist two independent integrals of motion, so the system is integrable, but their form is very complicated,

2. the only integral of motion is $E$ and other independent integrals are absent?

I'm not a specialist in dynamical systems and related complicated mathematics, so any simple explanations will be appreciated. I've found the related questions Idea of integrable systems and Non-integrability of the 2D double pendulum but didn't get any simple answer.

Answer 1

I think one must distinguish between chaotic "Hamiltonian" systems and chaotic dissipative systems. In the latter, the phase space volume is not conserved, so it is much more difficult to find "integrals of motion" because Liouville's theorem is broken. Remember, a quantity "A" is an integral of motion if

$\frac{dA}{dt} = \frac{\partial A}{\partial t} + \{A,H\} = 0$, where $H$ is the Hamiltonian. For dissipative chaotic systems, you can't even write down $H$, so it is difficult to see how one could generally find integrals/constants of motion of the system.

However, there is an important class of systems that show up in cosmology for example where you have Hamiltonian "chaos", where essentially the trajectories of the system exhibit all of the properties of chaos: sensitive dependence on initial conditions, diverging trajectories over time, but, the system still has attractors: a famous example is the dynamics of a closed anisotropic universe / Bianchi IX, in shameless self-promotion here: https://arxiv.org/pdf/1311.0389.pdf (in particular, see Page 27) This has of course led to wide debates for years in the cosmology community of whether this is "Really" chaos, since, in principle, the trajectories are predictable, but, I hope this answers your question.

Further, with respect to your Billiards problem / the famous Hadamard billiards, as you can see it is the same as the diagram on Page 27. Therefore, the billiard problem is also an example of Hamiltonian / deterministic / non-dissipative chaos. The phase space has an asymptotic attractor. This hopefully demonstrates that integrals of motion such as the one you found above ($E$ is the total energy of the system, and in this case, is the Hamiltonian, $H$) are only really possible if one can write down a Hamiltonian by Liouville's theorem.

Answer 2

I am not an expert on these issues, but if a further integral existed the orbit would be confined in a codimesion-1 embedded submanifold (for almost all the possible values of that function due to Sard's theorem). An embedded submanifold is a very regular subset, it cannot have self-intersections and cannot be dense in the space for example. Instead the orbit of a chaotic system does not seem to belong to such a regular set...However without a precise definition, all that remains just a suggestion...

Answer 3

The Bunimovich stadium is well known to be ergodic. Here is a nice description by Terry Tao. This notion extends naturally to quantum (or wave) chaos where instead of the trajectories being asymptotically uniformly distributed, it is the nodal domains of the eigenfunctions that are asymptotically uniform. The new book by Steve Zelditch "Eigenfunctions of the Laplacian on a Riemannian manifold" is a thorough technical look at these issues for the quantum/wave case. Free pdf here

The existence of scars (asymptotic concentrations in phase space) is a reflection of unstable periodic trajectories. If there are no concentrations then the system is called uniquely ergodic.

Integrability requires not just N constants of motion, but also that the constants be in involution to one another. This means that the Poisson Brackets of any pair is zero. By the way, there is a difference between constants of motion and integrals of motion. Integrals of motion comprise a subset of the constants of motion. Here is a reference to ergodicity vs integrability.

The term chaos is used to mean different things in different contexts. In addition to the spreading of trajectories you need mixing to get classical chaos. Thus you'll find extensive discussion of these issues in books on Riemannian Geometry since the mixing usually comes from boundary curvature. My favorite is Marcel Berger's A Panoramic View of Riemannian Geometry. Berger has an extensive discussion of stadium billiards.

Answer 4

1. On one hand, a positive maximal Lyapunov exponent (MLE) is often taken as a de-facto definition of (deterministic) chaos. (Note that chaos also requires topological mixing.)

2. On the other hand, Poincare showed that an autonomous Liouville-integrable Hamiltonian system has only zero Lyapunov exponents along periodic orbits, cf. e.g. my Phys.SE answer here. (If the level-sets are compact, then every orbit is periodic, cf. the Liouville-Arnold theorem. For details, see e.g. my Phys.SE answer here.)

Answer 5

This depends in ways on the system. Yet integrable domains can exist. The logistics map $x_{n+1}~=~rx_n(1~-~x_n)$ has for the parameter $r$ zones of stability. The image

illustrates bifurcation regions of regular dynamics. These are islands embedded in this region of "scarring" with chaotic dynamics.