Question

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what a non-integrable system is.) In particular, is there a dichotomy between "integrable" and "chaotic"? (There is an interesting wikipedia article but I don't find it completely satisfying.)

Update (Dec 2010): Thanks for the many excellent answers. I came across another quote from Nigel Hitchin:

"Integrability of a system of differential equations should manifest itself through some generally recognizable features:

• the existence of many conserved quantities

• the presence of algebraic geometry

• the ability to give explicit solutions.

These guidelines whould be interpreted in a very broad sense."

(If there are some aspects mentioned by Hitchin not addressed by the current answers, additions are welcome...)

Closely related questions: What does it mean for a differential equation "to be integrable"?; basic questions on quantum integrable systems

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Gil Kalai

Comments

Very good answers! I'd love to see more angles to this important issue, which is why a little bounty is offered.

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Gil Kalai

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Excellent question, I think. But I'm stuck before we get to the "integrable" part. What is a "system"? I'd be glad if someone addressed this in their answer.

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Tom Leinster

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I believe that 'system' is in the same sense as 'dynamical system', which probably comes from 'system of differential equations'.

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user José Figueroa-O'Farrill

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Thanks, José, but that doesn't really answer the question. People use "dynamical system" in a variety of ways. E.g. the wikipedia article en.wikipedia.org/wiki/Dynamical_system_(definition) gives the general definition as a partial action of a monoid on a set. An article by Adler in the Bulletin of the AMS defines it as a compact metric space with a continuous endomorphism. But I don't think that either of those definitions is what the answers below are referring to. Perhaps I should ask this as a separate question

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Tom Leinster

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The book by Hitchin, Segal, Ward and Woodhouse begins with this nice quote: "Integrable systems, what are they? It's not easy to answer precisely. The question can occupy a whole book (Zakharov 1991), or be dismissed as Louis Armstrong is reputed to have done once when asked what jazz was---'If you gotta ask, you'll never know!'"

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user HJRW

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Could anyone with enough rep please add the "integrable-systems" tag to this question?

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user mathphysicist

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There was a Gibbs lecture "Integrable Systems: A modern View": jointmathematicsmeetings.org/meetings/national/jmm/deift

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Thomas Riepe

Answer 1

I would like to add one more example of integrability which refers to Hopf algebras and is probably the easiest to formulate. It naturally arises in spin chain physics, but can be treated abstractly as well. Consider (semi-simple) Lie algebra $\mathfrak{g}$, its universal enveloping algebra $\mathfrak{h}=U(\mathfrak{g})$ and its Hopf double. The latter has coproduct homomorphism $\Delta: \mathfrak{g}\to \mathfrak{g}\otimes\mathfrak{g}$. Now let us consider an operator $\mathfrak{R}$ (so-called, R-matrix) as the following mapping $\mathfrak{R}:\mathfrak{h}\otimes\mathfrak{h}\to\mathfrak{h}\otimes\mathfrak{h}$, meaning that $\mathfrak{R}$ is some tensor product of polynomials of elements from $\mathfrak{g}$. The integrability condition reads

$[\Delta,\mathfrak{R}]=0,$

viz. the coproduct should commute with the R-matrix. It is now a matter of several lines of simple calculations to show that the Yang-Baxter equation on the R-matrix, which is frequently referred to as the necessary condition for integrability follows [see, e.g. Kassel "Quantum Gorups"].

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Peter Koroteev

Answer 2

For a Hamiltonian system, integrable means the solution lies on a surface (in phase space). Can anyone challenge that? The dimension of the surface depends on how many integrals there are. And as long as we are talking about Hamiltonian systems, chaotic and integrable are indeed complements of each other, but this is not the case for dynamical systems in general.

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Norbert S

Answer 3

"All integrable Hamiltonian systems are alike, while each nonintegrable one is nonintegrable in its own way", Valerij V. Kozlov (after L. N. Tolstoi of course), Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, 1996

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user J.J. Green

Answer 4

The integrability conditions for the existence of a Lagrangian or Hamiltonian are known as "conditions of variational self-adjointness."

The conditions are studied in the context of the Inverse Problem, which

• Ruggero Maria Santilli, Foundations of Theoretical Mechanics: The Inverse Problem in Newtonian Mechanics, vol. 1, 2 vols. (New York: Springer-Verlag, 1978), p. 10,

formulates as:

Given the totality of solutions $y(x) = \left\{y^1(x), \ldots, y^n(x)\right\}$ of a system of $n$ ordinary differential equations of order $r$, $$F_k\left(x, y^{(0)}, y^{(1)}, \ldots, y^{(r)}\right)=0\qquad\text{(I.23)}$$ $$y^{(i)}=\frac{d^iy}{dx^i},\qquad i=1,\ldots,r,\qquad k=1,2,\ldots,n,$$ determine whether there exists a functional $$A(y)=\int_{x_1}^{x_2}dxL\left(x,y^{(0)},\ldots,y^{(r-1)}\right)\qquad\text{(I.24)}$$ which admits such solutions as extremals.

It appears Helmholtz was the first to study the Inverse Problem (ibid., p. 12):

The necessary and sufficient conditions for the existence of a solution $L$ of system (I.29)* were apparently formulated for the first time by Helmholtz (1887)²⁶ on quite remarkable intuitional grounds. In essence, Helmholtz's starting point was the property of the self-adjointness of Lagrange's equations, i.e., their system of variational forms coincides with the adjoint system (see Chapter 2 and following). This is a property which goes back to Jacobi (1837).²⁷ Without providing a rigorous proof, Helmholtz indicated that the necessary and sufficient condition for the existence of a solution $L$ of system (I.28)** is that the system $F_k = 0$ be self-adjoint.

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<sup>26. Helmholtz did not consider an explicit dependence of the equations of motion on time. Subsequent studies indicated that his findings were insensitive to such a dependence.
27. The equations of variations of Lagrange's equations or, equivalently, of Euler's equations of a variational problem, are called Jacobi's equations in the current literature of the calculus of variations. We shall use the same terminology for our Newtonian analysis.</sup>

^{*(I.29) is the Euler-Lagrange equation corresponding to (I.24) when $n>1$, $r=2$.
**(I.28) is the case when $n=r=1$.}

This same analysis of conditions for variational self-adjointness of a Lagrangian can be applied to Hamiltonians, as Hamiltonians are simply the Legendre transform of Lagrangians (cf. Callen's Thermodynamics and an Introduction to Thermostatistics §5.2 (pp. 137-145) for a good introduction to Legendre transforms).

Helmholtz (1887) is:

• Ueber die physikalische Bedeutung des Prinzips der kleinsten Wirkung
  ("On the physical meaning of the principle of least action")
  J. Reine Angew. Math., 100 (1887), pp. 137–166
  • See esp. §3 "The derivation of the kinetic potential from the value of energy." (pp. 18-21 of the English translation), where he (as he says in his outline on p. 5) "treats the opposite problem, namely, that of deriving $H$ [the Hamiltonian] from $E$ [the energy]."

This post imported from StackExchange MathOverflow at 2018-08-28 16:20 (UTC), posted by SE-user Geremia