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?
[Physics] Chaos and integrability in classical mechanics
chaos theoryclassical-mechanicscomplex systemshamiltonian-formalismintegrable-systems
Related Solutions
(1) In general, what is meant by non-linear system in classical mechanics?
A linear system is described by a set of differential equations that are a linear combination of the dependent variable and its derivatives. Some examples of linear systems in classical mechanics:
- A damped harmonic oscillator, $$m \frac{d^2 x(t)}{dt^2} + c \frac{d x(t)}{dt} + k x(t) = 0$$
- The heat equation, $$\frac{\partial u(\vec x, t)}{\partial t} -\alpha \nabla^2 u(\vec x, t) = 0$$
- The wave equation, $$\frac{\partial^2 u(\vec x, t)}{\partial t^2} -c \nabla^2 u(\vec x, t) = 0$$
Non-linear systems cannot be described by a linear set of differential equations. Some examples of non-linear systems in classical mechanics:
- Aerodynamic drag, where the drag force is proportional to the square of velocity, $$F_d = \frac 1 2 \rho v^2 C_D A$$
- The Navier-Stokes equations, which are notoriously non-linear, $$\rho \left( \frac{\partial v}{\partial t} + \vec v \cdot \vec \nabla v \right) = -\vec \nabla p + \vec \nabla T + \vec f$$
- Gravitational systems, where the force is inversely proportional to the square of distance between objects, $$\vec F = -\frac {GMm}{||\vec r||^3}\vec r$$
(2) Furthermore, why is it that most non-linear systems are considered non-integrable?
That term, "non-integrable" has two very distinct meanings. One sense is that the integral cannot be expressed as a finite combination of elementary functions. The elementary functions are polynomials, rational roots, the exponential function, the logarithmic function, and the trigonometric functions. This is a rather arbitrary division. For example, the integrals $\operatorname{li}(x) = \int_0^x 1/\ln(t)\,dt$ and $\operatorname{Si}(x) = \int_0^x \sin(t)/t\,dt$ cannot be expressed in the elementary functions. These are the logarithmic and sine integrals. These "special functions" appear so often that algorithms have been devised to estimate their values. Categorizing functions as elementary versus non-elementary is a bit arbitrary.
Just because the solution to a problem can't be expressed in elementary functions doesn't mean the problem is unsolvable. It just mean it's not solvable in the elementary functions. For example, people oftentimes say the three body problem is not "solvable". That's nonsense (ignoring collision cases). In the sense of solvability in the elementary functions, even the two body problem is not "integrable". Kepler's equation, $M = E - e\sin E$, gets in the way. Just because the two body problem cannot be expressed in terms of a finite combination of elementary functions does not mean we can't solve the two body problem.
There's another sense of "integrability", which is "does the integral exist?" Going back to the n body problem, a problem exists with collisions. These collisions introduce singularities, so that one could say that the n body problem is not integrable in the case of collisions. Collisions represent one kind of singularity. Painlevé conjectured that the n body problem has collisionless singularities in when $n\ge 4$. This has been proven to be true when $n \ge 5$. Newtonian mechanics allows some configurations of gravitating point masses to be sent to infinity in finite time. This truly is an example of non-integrability.
Proving integrability (or lack thereof) in this sense is a much tougher problem than showing that a problem is (or is not) solvable in the elementary functions. There's a million dollar prize for the first person who can either prove that the Navier-Stokes equations have globally-defined, smooth solutions, or come up with a counterexample that shows that that the Navier-Stokes equations are not "integrable."
In this answer, we elaborate on the various definitions of integrability, separability & AA-property, in order to expose their (slight) differences.
Let there be given a finite-dimensional autonomous Hamiltonian system, defined on a connected $2n$-dimensional symplectic manifold $({\cal M},\{\cdot ,\cdot \})$.
Definition. The system is (completely) Liouville integrable if there exist $n$ functionally independent, Poisson-commuting, globally defined functions $F_1, \ldots, F_n: {\cal M}\to \mathbb{R}$, so that the Hamiltonian $H=H(F)$ is a function of $F_1, \ldots, F_n$, only. See also this related Phys.SE post.
Definition. The system is (completely) $H$-separable if there exists an atlas of Darboux coordinates $ q^1, \ldots, q^n, p_1, \ldots, p_n : {\cal U}\to \mathbb{R}$ with separation functions $F_1, \ldots, F_n: {\cal U}\to \mathbb{R}$ on triangular form $$F_1~=~F_1(q^1,p_1), \qquad F_2~=~F_2(q^2,p_2; F_1), \qquad F_3~=~F_3(q^3,p_3; F_1,F_2), \tag{1}$$ $$\qquad \ldots, \qquad F_n~=~F_n(q^n,p_n; F_1,\ldots, F_{n-1}), $$
such that the Hamiltonian $H=H(F)$ is a function of $F_1, \ldots, F_n$, only.Note that the separation functions $F_1, \ldots, F_n$ from Definition 3 are automatically Poisson-commuting and constants of motion, but not necessarily functionally independent. A globally defined $H$-separating Darboux coordinate system with functionally independent separation functions implies integrability.
Theorem. Integrability $\Rightarrow$ $H$-separability. Proof: Use Caratheodory-Jacobi-Lie theorem to extend the Poisson-commuting coordinates $(F_1, \ldots, F_n)$ into an atlas of Darboux coordinate neighborhoods. The Hamiltonian $H(F)$ is then on separable form. $\Box$
Definition. The system is called (completely) $W$-separable if there exists an atlas of Darboux coordinates $ q^1, \ldots, q^n, p_1, \ldots, p_n : {\cal U}\to \mathbb{R}$ and a Hamilton's characteristic function $W: {\cal U}\times \mathbb{R}^n\to \mathbb{R}$ of the form $$ W(q;\alpha)~=~ \sum_{k=1}^n W_k(q^k;\alpha_1, \ldots, \alpha_n),\tag{2}$$ where $\alpha=(\alpha_1,\ldots,\alpha_n)$ are $n$ independent integration constants, and where $$ p_k~:=~\frac{\partial W}{\partial q^k}, \qquad k~\in~\{1, \ldots, n\}. \tag{3}$$ such that the Hamilton-Jacobi (HJ) equation $$ H\left(q,\frac{\partial W(q;\alpha)}{\partial q}\right)~=~h(\alpha)\tag{4} $$ is satisfied. Here $h:\mathbb{R}^n\to \mathbb{R}$ is a given function.
Case where $W$-separability $\Rightarrow$ $H$-separability: Assume that the $n$ integration constants $\alpha=(\alpha_1,\ldots,\alpha_n)$ can be identified with Poisson-commuting separation functions $F_k(z)$, $k\in\{1, \ldots, n\}$. Then $H=h(F)$ and the separation functions become constants of motion.
$H$-separability does not necessarily imply $W$-separability as there is no guarantee that a globally defined Hamilton's characteristic function $W$ exists as a solution to the HJ equation.
Definition. The system has the AA-property if there exists an atlas of angle-action coordinates $(w^1,\ldots, w^n,J_1,\ldots, J_n)$, where the symplectic $2$-form $\omega=\sum_{k=1}^n\mathrm{d}J_k\wedge \mathrm{d}w^k$ is on Darboux form, where each AA-coordinate system is $w$-complete, and where the Hamiltonian $H=H(J)$ does not depend on the angles $w^k$. (We allow non-compact "angle" variables. The compact angle variables has unit period $w^k\sim w^k+1$.)
The AA-property clearly implies all the separability conditions. A globally defined angle-action coordinate system implies integrability.
Case where integrability $\Rightarrow$ AA-property: Assume $\forall f=(f_1,\ldots, f_n)\in\mathbb{R}^n$ that the level sets $\bigcap_{k=1}^n F_k^{-1}(\{f_k\})$ are compact in ${\cal M}$. Then the Liouville-Arnold theorem shows the AA-property. For a proof of the Liouville-Arnold theorem, see my Phys.SE answer here.
Best Answer
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.