Both books have extensive bibliographies with further references to look into. Now, as for classification and identification of new integrable systems of PDEs, at least in two independent variables, it turns out that the infinitesimal higher symmetries play an important role here. A recent collective monograph Integrability , edited by A.

## Contact geometry

Mikhailov and published by Springer in , could be a good starting point in this direction. See also another recent book Algebraic theory of differential equations edited by MacCallum and Mikhailov and published by Cambridge University Press. For a general introduction to the subject of symmetries of systems of PDEs, I can recommend the book Applications of Lie groups to differential equations by Peter Olver. This is soft -- but I think of an integrable system as one whose dynamics are dominated by algebra. For finite dimensional integrable systems, the symmetries related to conserved quantities by Noether's theorem force the trajectories to live on half-dimensional tori.

For infinite dimensional integrable systems, where the flow on the scattering data is isospectral the symmetries force solutions to be n-soliton solutions plus dispersive modes.

## Integrability and non-integrability in Hamiltonian mechanics

There is a blog post of Terry Tao's apologies for not having the link which talks about how algebra is the right tool to understand structure while analysis is the right tool to understand randomness. The claim is that one mark of an good problem is the presence of an interesting relationship between structure and randomness and hence the requirement that both algebra and analysis be used -- to some degree -- in order to get a good answer to the problem.

The soliton resolution conjecture is by this standard a good problem because the asymptotic n-soliton solutions are fundamentally algebraic while the dispersive modes are fundamentally analytic objects. I agree with Dmitri that there isn't a dichotomy. The symmetries can have a large or small role in the dynamics as can the ergodicity.

I'll give a bit of a physics definition. So, literally, an integrable system in this view is one that can be solved by a sequence of integrals which may not be explicitly solvable in elementary functions, of course. To connect to other answers, this should only work out when there are enough symmetries for us to write down and integrate. 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. The integrability condition reads. 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.

Kassel "Quantum Gorups"]. After reading several books and articles about integrable systems, and after several years of work in the field, I consider particularly meaningful the following quotation from Frederic Helein's book 'Constant mean curvature surfaces, harmonic maps and integrable systems', Lectures in Mathematics, ETH Zurich, Birkhauser Basel :. It will turn out that this contemplation is fruitful and lead to many results".

I will only talk about the finite dimensional case.

Now, the cool thing is that there exist action angle coordinates. As a possible application, KAM theory is usually formulated as an application to systems in action angle coordinates. This in turn implies that integrable systems are stable in a subtle measure theoretic sense.

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We have a great form of perturbation theory for integrable systems. I have found an article "Quantum signatures of an integrable system with a chaotic scattering map" here:. For a Hamiltonian system, integrable means the solution lies on a surface in phase space. 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.

Kozlov after L. The integrability conditions for the existence of a Lagrangian or Hamiltonian are known as "conditions of variational self-adjointness. It appears Helmholtz was the first to study the Inverse Problem ibid. In essence, Helmholtz's starting point was the property of the self-adjointness of Lagrange's equations, i. This is a property which goes back to Jacobi Helmholtz did not consider an explicit dependence of the equations of motion on time.

## lamofirmsicktinc.gqatical physics - What is an integrable system - MathOverflow

Subsequent studies indicated that his findings were insensitive to such a dependence. 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.

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. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What is an integrable system Ask Question. Asked 9 years, 10 months ago. Active 5 days ago. Viewed 27k times. 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. Gil Kalai Gil Kalai 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. It's not easy to answer precisely. The question can occupy a whole book Zakharov , 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!

The Newtonian planar three body problem, for most masses, has been proven to be non-integrable. Marius Kempe 20 20 silver badges 32 32 bronze badges. Richard Montgomery Richard Montgomery 6, 21 21 silver badges 42 42 bronze badges. Here is a little bit about what the Poisson bracket of two functions is that explains its meaning and why two functions with vanishing Poisson bracket are said to "Poisson commute".

Thus two functions Poisson commute iff the vector fields corresponding to their differentials commute, i. Ivan Izmestiev 4, 15 15 silver badges 40 40 bronze badges. Dick Palais Dick Palais Dmitri Panov Dmitri Panov In fact there is a memoir of the AMS by Markus and Meyer which shows that a generic Hamiltonian system is neither integrable nor ergodic, see books.

Paris, pp. Together with two preceding notes in the same journal, this generalises the concept of "complete integrability" of a mechanical system. Aaron Hoffman Aaron Hoffman 1 1 gold badge 4 4 silver badges 8 8 bronze badges.

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