Integrable system
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In the context of differential equations to integrate an equation means to solve it from initial conditions. Accordingly, an integrable system is a system of differential equations whose behavior is determined by initial conditions and which can be integrated from those initial conditions.
Many systems of differential equations arising in physics are integrable. A standard example is the motion of a rigid body about its center of mass. This system gives rise to a number of conserved quantities: the angular momenta. Conserved quantities such as these are also known as the first integrals of the system. Roughly speaking, if there are enough first integrals to give a coordinate system on the set of solutions, then it is possible to reduce the original system of differential equations to an equation that can be solved by computing an explicit integral. Other examples giving rise to integrable systems in physics are some models of shallow water waves (Korteweg–de Vries equation), the nonlinear Schrödinger equation, and the Toda lattice in statistical mechanics.
While the presence of many conserved quantities is generally a fairly clearcut criterion for integrability, there are other ways in which integrability can appear. It is famously difficult to be precise about what the term means. Nigel Hitchin (Hitchin, Segal & Ward, p. 1) draws a comparison to a quotation by jazz musician Louis Armstrong, who when asked what jazz was, is rumored to have replied "If you gotta ask, you'll never know." Hitchin identifies three generally recognizable features of integrable systems:
 the existence of many conserved quantities
 the presence of algebraic geometry
 the ability to give explicit solutions
Contents
 1 Frobenius integrability (overdetermined differential systems)
 2 General dynamical systems
 3 Hamiltonian systems and Liouville integrability
 4 Actionangle variables
 5 The Hamilton–Jacobi approach
 6 Solitons and inverse spectral methods
 7 Quantum integrable systems
 8 Exactly solvable models
 9 List of some wellknown classical integrable systems
 10 See also
 11 Notes
 12 References
 13 Further reading
 14 External links
Frobenius integrability (overdetermined differential systems)
A differential system is said to be completely integrable in the Frobenius sense if the space on which it is defined has a foliation by maximal integral manifolds. The Frobenius theorem states that a system is completely integrable if and only if it generates an ideal that is closed under exterior differentiation. (See the article on integrability conditions for differential systems for a detailed discussion of foliations by maximal integral manifolds.)
General dynamical systems
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 submanifolds of the smallest possible dimension that are invariant under the flow. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville (see below), which is what is most frequently referred to in this context.
An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of differential equations or finite difference equations.
The distinction between integrable and nonintegrable dynamical systems thus has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in exact form.
Hamiltonian systems and Liouville integrability
In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense, see the LiouvilleArnold theorem. Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated to the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of Poisson commuting invariants (i.e., functions on the phase space whose Poisson brackets with the Hamiltonian of the system, and with each other, vanish).
In finite dimensions, if the phase space is symplectic (i.e., the center of the Poisson algebra consists only of constants), then it must have even dimension , and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is . The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is called Lagrangian. All autonomous Hamiltonian systems (i.e. those for which the Hamiltonian and Poisson brackets are not explicitly time dependent) 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 leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical form are called the action variables, and the resulting canonical coordinates are called actionangle variables (see below).
There is also a distinction between complete integrability, in the Liouville sense, and partial integrability, as well as a notion of superintegrability and maximal superintegrability. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is superintegrable. If there is a regular foliation with onedimensional leaves (curves), this is called maximally superintegrable.
Actionangle variables
When a finitedimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as actionangle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the torus. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
The Hamilton–Jacobi approach
In canonical transformation theory, there is the 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 actionangle variables. In the general theory of partial differential equations of 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.
Solitons and inverse spectral methods
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that solitons, which are strongly stable, localized solutions of partial differential equations like the Korteweg–de Vries equation (which describes 1dimensional nondissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinitedimensional integrable 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 problems), which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.
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, cf. Lax pair. 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 actionangle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.
Quantum integrable systems
There is also a notion of quantum integrable systems.
In the quantum setting, functions on phase space must be replaced by selfadjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators.
To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are onebody reducible. A quantum system is said to be integrable if the dynamics are twobody reducible. The YangBaxter equation is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the quantum inverse scattering method where the algebraic Bethe Ansatz can be used to obtain explicit solutions. Examples of quantum integrable models are the LiebLiniger Model, the Hubbard model and several variations on the Heisenberg model.^{[1]} Some other types of quantum integrability are known in explicitly timedependent quantum problems, such as the driven TavisCummings model ^{[2]}.
Exactly solvable models
In physics, completely integrable systems, especially in the infinitedimensional 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.
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 equations 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.
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.^{[citation needed]}
List of some wellknown classical integrable systems
1. Classical mechanical systems (finitedimensional phase space):
 Harmonic oscillators in n dimensions
 Central force motion (exact solutions of classical centralforce problems)
 Two center Newtonian gravitational motion
 Geodesic motion on ellipsoids
 Neumann oscillator
 Lagrange, Euler, and Kovalevskaya tops
 Integrable Clebsch and Steklov systems in fluids
 Calogero–Moser–Sutherland model^{[3]}
 Swinging Atwood's Machine with certain choices of parameters
2. Integrable lattice models
 Toda lattice
 Ablowitz–Ladik lattice
 Volterra lattice
 Multistate Landau–Zener Models^{[4]}
3. Integrable systems of PDEs in 1 + 1 dimension
 Korteweg–de Vries equation
 Sine–Gordon equation
 Nonlinear Schrödinger equation
 AKNS system
 Boussinesq equation (water waves)
 CamassaHolm equation
 Nonlinear sigma models
 Classical Heisenberg ferromagnet model (spin chain)
 Classical Gaudin spin system (Garnier system)
 KaupKupershmidt equation
 KricheverNovikov equation
 Landau–Lifshitz equation (continuous spin field)
 Benjamin–Ono equation
 DegasperisProcesi equation
 Dym equation
 Threewave equation
4. Integrable PDEs in 2 + 1 dimensions
 Kadomtsev–Petviashvili equation
 Davey–Stewartson equation
 Ishimori equation
 NovikovVeselov equation
5. Other integrable systems of PDEs in higher dimensions
 Donaldson theory
 Systems with contact Lax pairs^{[5]}
See also
Notes
 ^ V.E. Korepin, N. M. Bogoliubov, A. G. Izergin (1997). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. ISBN 9780521586467.
 ^ N. A. Sinitsyn; F. Li (2016). "Solvable multistate model of LandauZener transitions in cavity QED". Phys. Rev. A. 93 (6): 063859. arXiv:1602.03136. Bibcode:2016PhRvA..93f3859S. doi:10.1103/PhysRevA.93.063859.
 ^ F. Calogero (2008) CalogeroMoser system. Scholarpedia, 3(8):7216.
 ^ N. A. Sinitsyn; V. Y. Chernyak (2017). "The Quest for Solvable Multistate LandauZener Models". arXiv:1701.01870 [quantph].
 ^ A. Sergyeyev (2018). New integrable (3+1)dimensional systems and contact geometry, Lett. Math. Phys. 108, no. 2, 359376, arXiv:1401.2122 doi:10.1007/s1100501710134
References
 V. I. Arnold (1997). Mathematical Methods of Classical Mechanics, 2nd ed. Springer. ISBN 9780387968902.
 O. Babelon, D. Bernard, M. Talon (2003). Introduction to classical integrable systems. Cambridge University Press. doi:10.1017/CBO9780511535024. ISBN 052182267X.
 R.J. Baxter (1982). Exactly solved models in statistical mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. ISBN 9780120831807.
 M. Dunajski (2009). Solitons, Instantons and Twistors,. Oxford University Press. ISBN 9780198570639.
 L. D. Faddeev, L. A. Takhtajan (1987). Hamiltonian Methods in the Theory of Solitons. AddisonWesley. ISBN 9780387155791.
 A. T. Fomenko, Symplectic Geometry. Methods and Applications. Gordon and Breach, 1988. Second edition 1995, ISBN 9782881249013.
 A. T. Fomenko, A. V. Bolsinov Integrable Hamiltonian Systems: Geometry, Topology, Classification. Taylor and Francis, 2003, ISBN 9780415298056.
 H. Goldstein (1980). Classical Mechanics, 2nd. ed. AddisonWesley. ISBN 0201029189.
 J. Harnad, P. Winternitz, G. Sabidussi, eds. (2000). Integrable Systems: From Classical to Quantum. American Mathematical Society. ISBN 0821820931.
 J. Hietarinta, N. Joshi, F. Nijhoff (2016). Discrete systems and integrability. Cambridge University Press. doi:10.1017/CBO9781107337411. ISBN 9781107042728.
 N Hitchin; G Segal; R Ward (1999), Integrable systems: Twistors, loop groups, and Riemann surfaces, Clarendon Press
 V. E. Korepin, N. M. Bogoliubov, A. G. Izergin (1997). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. ISBN 9780521586467.
 V. S. Afrajmovich, V. I. Arnold, Yu. S. Il'yashenko, L. P. Shil'nikov. Dynamical Systems V. Springer. ISBN 3540181733.
 Giuseppe Mussardo. Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics. Oxford University Press. ISBN 9780199547586.
 G. Sardanashvily (2015). Handbook of Integrable Hamiltonian Systems. URSS. ISBN 9785396006874.
Further reading
 A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves [1]
 Donagi, R., Markman, E., Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles, in “Integrable systems and quantum groups” (Montecatini Terme, 1993), Lecture Notes in Math., 1620, pp. 1119, Springer, Berlin, 1996.
 Michèle Audin, Spinning Tops: A Course on Integrable Systems, Cambridge University Press
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "Integrable system", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104