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| In [[mathematics]] and [[physics]], there are various distinct notions that are referred to under the name of '''integrable systems'''.
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| In the general theory of differential systems, there is ''[[#Frobenius integrability (overdetermined differential systems)|Frobenius integrability]]'', which refers to overdetermined systems. In the classical theory of [[Hamiltonian mechanics|Hamiltonian]] [[dynamical systems]], there is the notion of ''[[#Hamiltonian_systems_and_Liouville_integrability|Liouville integrability]]''. More generally, in differentiable dynamical systems integrability relates to the existence of [[foliation]]s by invariant submanifolds within the [[phase space]]. Each of these notions involves an application of the idea of foliations, but they do not coincide. There are also notions of ''complete integrability'', or ''exact solvability'' in the setting of quantum systems and statistical mechanical models. Integrability can often be traced back to the [[algebraic geometry]] of [[differential operator]]s.
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| ==Frobenius integrability (overdetermined differential systems)==
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| A differential system is said to be ''completely integrable'' in the [[Ferdinand Georg Frobenius|Frobenius]] sense if the space on which it is defined has a [[foliation]] by maximal [[integral manifold]]s. The [[Frobenius theorem (differential topology)|Frobenius theorem]] states that a system is completely integrable if and only if it generates an ideal that is closed under [[exterior derivative|exterior differentiation]]. (See the article on [[integrability conditions for differential systems]] for a detailed discussion of foliations by maximal integral manifolds.)
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| ==General dynamical systems==
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| In the context of differentiable [[dynamical systems]], the notion of '''integrability''' refers to the existence of invariant, regular foliations; i.e., ones whose leaves are [[embedded submanifold]]s of the smallest possible dimension that are invariant under the [[Flow (mathematics)|flow]]. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation.
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| This concept has a refinement in the case of [[Hamiltonian mechanics|Hamiltonian systems]], known as '''complete integrability in the sense of [[Liouville]]''' (see below), which is what is most frequently referred to in this context.
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| An extension of the notion of integrability is also applicable to discrete systems such as lattices.
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| This definition can be adapted to describe '''evolution equations''' that either are systems of
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| [[differential equations]] or [[finite difference| finite difference equations]].
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| The distinction between integrable and nonintegrable dynamical systems thus has the qualitative
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| implication of '''regular motion''' vs. [[chaotic motion]] and hence is an intrinsic property, not just a matter of whether
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| a system can be explicitly integrated in exact form.
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| ==Hamiltonian systems and Liouville integrability==
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| In the special setting of [[Hamilton's equations|Hamiltonian systems]], we have the notion of integrability in the [[Liouville]] sense.
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| '''Liouville integrability''' means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields
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| associated to the invariants of the foliation span the tangent distribution. Another way to state this is that there exists
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| a '''maximal set''' of Poisson commuting invariants (i.e., functions on the phase space whose [[Poisson bracket]]s with the Hamiltonian of the system,
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| and with each other, vanish).
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| In finite dimensions, if the [[phase space]] is [[Symplectic geometry|symplectic]] (i.e., the center of the Poisson algebra consists only of constants), then it must have
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| even dimension <math>2n </math>, and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is
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| <math>n </math>. The leaves of the foliation are [[Isotropic quadratic form|totally isotropic]] with respect to the symplectic form and such a maximal isotropic foliation is
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| called '''[[Lagrangian submanifold|Lagrangian]]'''. All ''autonomous'' Hamiltonian systems (i.e. those for which the Hamiltonian and Poisson brackets are not explicitly time dependent)
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| have at least one invariant; namely, the '''Hamiltonian''' itself, whose value along the flow is the energy. If the energy level sets are compact, the
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| leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical <math> 1 </math>-form
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| are called the '''action''' variables, and the resulting canonical coordinates are called '''[[action-angle variables]]''' (see below).
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| There is also a distinction between '''complete integrability''', in the [[Liouville]] sense, and '''partial integrability''', as well as
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| a notion of '''[[Superintegrable Hamiltonian system|superintegrability]]''' and '''maximal superintegrability'''. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation.
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| When the number of independent Poisson commuting invariants is less than maximal (but, in the case of
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| autonomous systems, more than one), we say the system is '''partially integrable'''.
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| When there exist further functionally independent invariants, beyond the maximal number that
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| can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is
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| less than n, we say the system is '''superintegrable'''. If there is a regular foliation with one-dimensional
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| leaves (curves), this is called '''maximally superintegrable'''.
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| ==Action-angle variables==
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| When a finite dimensional Hamiltonian system is completely integrable in the Liouville sense,
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| and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are [[torus|tori]].
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| There then exist, as mentioned above, special sets of [[canonical coordinates]] on the [[phase space]] known as [[action-angle variables]],
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| such that the invariant tori are the joint level sets of the [[action (physics)|action]] variables. These thus provide a complete set of invariants
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| of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the
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| invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
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| == The Hamilton–Jacobi approach ==
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| In [[canonical transformation]] theory, there is the [[Hamilton–Jacobi equations|Hamilton–Jacobi method]], in which solutions to Hamilton's equations are sought by first finding a '''complete solution''' of the associated [[Hamilton–Jacobi equation]]. In classical terminology, this is described as determining a transformation to a canonical set of coordinates consisting of '''completely ignorable variables'''; i.e., those in which there is no dependence of the Hamiltonian on a complete set of canonical "position" coordinates, and hence the corresponding canonically conjugate momenta are all conserved quantities. In the case of compact energy level sets, this is the first step towards determining the [[action-angle variables]]. In the general theory of partial differential equations of [[Hamilton–Jacobi equations|Hamilton–Jacobi]] type, a '''complete solution''' (i.e. one that depends on ''n'' independent constants of integration, where ''n'' is the dimension of the configuration space), exists in very general cases, but only in the local sense. Therefore the existence of a complete solution of the [[Hamilton–Jacobi equation]] is by no means a characterization of complete integrability in the Liouville sense. Most cases that can be "explicitly integrated" involve a complete [[separation of variables]], in which the separation constants provide the complete set of integration constants that are required. Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.
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| ==Solitons and inverse spectral methods==
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| A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that [[soliton]]s, which are strongly stable, localized solutions of partial differential equations like the [[Korteweg–de Vries equation]] (which describes 1-dimensional non-dissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinite dimensional integrable
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| Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the [[inverse scattering transform]] and more general '''inverse spectral methods''' (often reducible to [[Riemann–Hilbert problem]]s),
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| which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.
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| The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution. This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of '''Liouville integrability'''. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to '''completely ignorable coordinates''', in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.
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| ==Quantum integrable systems==
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| There is also a notion of '''quantum integrable systems'''.
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| In the quantum setting, functions on phase space must be replaced by [[self-adjoint operators]] on a [[Hilbert space]], and the notion
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| of '''Poisson commuting functions''' replaced by '''commuting operators'''.
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| To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum
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| system is said to be integrable if the dynamics are two-body irreducible. The [[Yang-Baxter equation]] is a consequence of this reducibility and leads to
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| trace identities which provide an infinite set of conserved quantities. All of this ideas are incorporated into the [[Quantum inverse scattering method]] where the
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| algebraic [[Bethe Ansatz]] can be used to obtain explicit solutions. Examples of quantum integrable models are the [[Lieb-Liniger Model]], the [[Hubbard model]] and
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| several variations on the [[Heisenberg model (quantum)|Heisenberg model]]. <ref>{{cite book | author=[[Vladimir Korepin|V.E. Korepin]], N. M. Bogoliubov, A. G. Izergin | title=Quantum Inverse Scattering Method and Correlation Functions| publisher=Cambridge University Press | year = 1997 | isbn=978-0-521-58646-7}}</ref>
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| ==Exactly solvable models==
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| In physics, completely integrable systems, especially in the infinite dimensional setting, are often referred to as '''exactly solvable models'''. This obscures the distinction between integrability in the Hamiltonian sense, and the more general dynamical systems sense.
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| There are also '''exactly solvable models''' in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the [[Bethe ansatz]] approach, in its modern sense, based on the '''[[Yang–Baxter equation]]s''' and the '''[[quantum inverse scattering method]]''' provide quantum analogs of the '''inverse spectral methods'''. These are equally important in the study of solvable models in statistical mechanics.
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| An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems.
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| ==List of some well-known classical integrable systems== | |
| '''1. Classical mechanical systems (finite-dimensional phase space):'''
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| * [[Harmonic oscillator]]s in ''n'' dimensions
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| * [[Central force]] motion ([[exact solutions of classical central-force problems]])
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| * Two center [[Newtonian gravitation]]al motion
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| * [[Geodesic motion on ellipsoids]]
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| * [[Neumann oscillator]]
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| * [[Lagrange, Euler and Kovalevskaya tops]]
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| * [[Integrable Clebsch and Steklov systems in fluids]]
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| * [[Calogero–Moser–Sutherland model]]
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| * [[Swinging Atwood's Machine]] with certain choices of parameters
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| '''2. Integrable lattice models'''
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| * [[Toda lattice]]
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| * [[Ablowitz–Ladik lattice]]
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| * [[Volterra lattice]]
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| '''3. Integrable systems of PDEs in 1 + 1 dimension'''
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| * [[Korteweg–de Vries equation]]
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| * [[Sine–Gordon equation]]
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| * [[Nonlinear Schrödinger equation]]
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| * [[AKNS system]]
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| * [[Boussinesq equation (water waves)]]
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| * [[Nonlinear sigma models]]
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| * [[Classical Heisenberg ferromagnet model (spin chain)]]
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| * [[Classical Gaudin spin system (Garnier system)]]
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| * [[Landau–Lifshitz equation (continuous spin field)]]
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| * [[Benjamin–Ono equation]]
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| * [[Dym equation]]
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| * [[Three-wave equation]]
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| '''4. Integrable PDEs in 2 + 1 dimensions'''
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| * [[Kadomtsev–Petviashvili equation]]
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| * [[Davey–Stewartson equation]]
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| * [[Ishimori equation]]
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| '''5. Other integrable systems of PDEs in higher dimensions'''
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| * [[Self-dual Yang–Mills equations]]
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| ==Notes==
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| <references/>
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| ==References==
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| * {{cite book | author=[[Vladimir Arnold|V.I. Arnold]] | title=Mathematical Methods of Classical Mechanics, 2nd ed. | publisher=Springer | year = 1997|isbn=978-0-387-96890-2 }}
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| * {{cite book | author=[[Maciej Dunajski|M. Dunajski]] | title=Solitons, Instantons and Twistors,| publisher=Oxford University Press| year = 2009|isbn=978-0-19-857063-9 }}
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| * {{cite book | author=[[Ludvig Faddeev|L.D. Faddeev]], L. A. Takhtajan | title =Hamiltonian Methods in the Theory of Solitons | publisher= Addison-Wesley | year=1987 |isbn=978-0-387-15579-1 }}
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| * [[Anatoly Fomenko|A.T. Fomenko]], ''Symplectic Geometry. Methods and Applications.'' Gordon and Breach, 1988. Second edition 1995, ISBN 978-2-88124-901-3.
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| * [[Anatoly Fomenko|A.T. Fomenko]], A. V. Bolsinov ''Integrable Hamiltonian Systems: Geometry, Topology, Classification''. Taylor and Francis, 2003, ISBN 978-0-415-29805-6.
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| * {{cite book | author=[[Herbert Goldstein|H. Goldstein]] | title =Classical Mechanics, 2nd. ed. | publisher= Addison-Wesley | year=1980 |isbn= 0-201-02918-9}}
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| * {{cite book | author=[[John Harnad|J. Harnad]], P. Winternitz, [[Gert Sabidussi|G. Sabidussi]], eds. | title=Integrable Systems: From Classical to Quantum| publisher=American Mathematical Society | year = 2000 | isbn=0-8218-2093-1 }}
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| * {{cite book | author=[[Vladimir Korepin|V.E. Korepin]], N. M. Bogoliubov, A. G. Izergin | title=Quantum Inverse Scattering Method and Correlation Functions| publisher=Cambridge University Press | year = 1997 | isbn=978-0-521-58646-7}}
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| * {{cite book | author=[[Valentin Afraimovich|V. S. Afrajmovich]], V.I. Arnold, Yu S. Il'yashenko, L. P. Shil'nikov | title=Dynamical Systems V | publisher=Springer | isbn=3-540-18173-3 }}
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| * {{cite book | author=Giuseppe Mussardo | title=Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics | publisher=Oxford University Press | ISBN=978-0-19-954758-6}}
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| ==External links==
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| * {{springer|title=Integrable system|id=p/i051330}}
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| {{DEFAULTSORT:Integrable System}}
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| [[Category:Dynamical systems]]
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| [[Category:Hamiltonian mechanics]]
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| [[Category:Integrable systems]]
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| [[Category:Partial differential equations]]
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