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In mathematics, integrability is a property of certain
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or
first integral In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...
s, such that its behaviour has far fewer degrees of freedom than the dimensionality of its
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
(a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ''generic'' dynamical systems, which are more typically
chaotic systems Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time. Many systems studied in physics are completely integrable, in particular, in the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two. Other elementary examples include the motion of a rigid body about its center of mass (the
Euler top In classical mechanics, the precession of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, ...
) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the
Lagrange top In classical mechanics, the precession of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, ...
). The modern theory of integrable systems was revived with the numerical discovery of solitons by
Martin Kruskal Martin David Kruskal (; September 28, 1925 – December 26, 2006) was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and ...
and
Norman Zabusky Norman J. Zabusky was an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work in ...
in 1965, which led to the
inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
method in 1967. It was realized that there are ''completely integrable'' systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves ( Korteweg–de Vries equation), the Kerr effect in optical fibres, described by the
nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in non ...
, and certain integrable many-body systems, such as the Toda lattice. In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this implies the Liouville-Arnold theorem; i.e., the existence of
action-angle variables In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving ...
. General dynamical systems have no such conserved quantities; in the case of autonomous
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic. A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution) if, locally, it has a
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
by maximal integral manifolds. But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds. Integrable systems do not necessarily have solutions that can be expressed in closed form or in terms of special functions; in the present sense, integrability is a property of the geometry or topology of the system's solutions in phase space.


General dynamical systems

In the context of differentiable dynamical systems, the notion of integrability refers to the existence of invariant, regular
foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
s; i.e., ones whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the
flow Flow may refer to: Science and technology * Fluid flow, the motion of a gas or liquid * Flow (geomorphology), a type of mass wasting or slope movement in geomorphology * Flow (mathematics), a group action of the real numbers on a set * Flow (psyc ...
. 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 has the qualitative implication of regular motion vs.
chaotic motion Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have ...
and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated into an 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 Liouville–Arnold theorem.) Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated with 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 bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
s with the Hamiltonian of the system, and with each other, vanish). In finite dimensions, if the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
is symplectic (i.e., the center of the Poisson algebra consists only of constants), it must have even dimension 2n , and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is n . 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 1 -form are called the action variables, and the resulting canonical coordinates are called
action-angle variables In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving ...
(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 one-dimensional leaves (curves), this is called maximally superintegrable.


Action-angle variables

When a finite-dimensional 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 In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
on the
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
known as
action-angle variables In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving ...
, such that the invariant tori are the joint level sets of the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
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 In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canoni ...
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
action-angle variables In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving ...
. 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 mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
, 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
soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
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 Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the
inverse scattering transform In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to sol ...
and more general inverse spectral methods (often reducible to
Riemann–Hilbert problem In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems ...
s), 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 action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.


Hirota bilinear equations and τ-functions

Another viewpoint that arose in the modern theory of integrable systems originated in a calculational approach pioneered by ''Ryogo Hirota'', which involved replacing the original nonlinear dynamical system with a bilinear system of constant coefficient equations for an auxiliary quantity, which later came to be known as the \tau-function. These are now referred to as ''Hirota equations''. Although originally appearing just as a calculational device, without any clear relation to the
inverse scattering In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, wh ...
approach, or the Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such as solitons could be derived. Subsequently, this was beautifully interpreted, by Mikio Sato and his students, at first for the case of integrable hierarchies of PDE's, such as the Kadomtsev-Petviashvili hierarchy, but then for much more general classes of integrable hierarchies, as a sort of ''universal phase space'' approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abelian group action on a (finite or infinite) Grassmann manifold. The \tau-function was viewed as the determinant of a projection operator from elements of the group orbit to some ''origin'' within the Grassmannian, and the ''Hirota equations'' as expressing the Plucker relations, characterizing the Plücker embedding of the Grassmannian in the projectivizatin of a suitably defined (infinite) exterior space, viewed as a fermionic Fock space.


Quantum integrable systems

There is also a notion of quantum integrable systems. In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized to
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administra ...
conservation laws. Every
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
has an infinite set of conserved quantities given by projectors to its energy
eigenstates In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
. However, this does not imply any special dynamical structure. To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body reducible. The
Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve the ...
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
Lieb–Liniger model The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. Introduction A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in ...
, the
Hubbard model The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard. The Hubbard model states that each el ...
and several variations on the Heisenberg model. Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model.


Exactly solvable models

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. 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 In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve the ...
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. 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.


List of some well-known integrable systems

;Classical mechanical systems (finite-dimensional phase space) * Harmonic oscillators in ''n'' dimensions *
Central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...
motion ( exact solutions of classical central-force problems) * Two center
Newtonian gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
al motion * Geodesic motion on ellipsoids * Neumann oscillator *
Lagrange, Euler, and Kovalevskaya tops In classical mechanics, the precession of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, ...
* Integrable Clebsch and Steklov systems in fluids * Calogero–Moser–Sutherland model ;Integrable lattice models * Toda lattice * Ablowitz–Ladik lattice * Volterra lattice ;Integrable systems in 1 + 1 dimensions * Korteweg–de Vries equation * Sine–Gordon equation *
Nonlinear Schrödinger equation In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in non ...
* AKNS system *
Boussinesq equation (water waves) In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by ...
* Camassa-Holm equation *
Nonlinear sigma models In quantum field theory, a nonlinear ''σ'' model describes a scalar field which takes on values in a nonlinear manifold called the target manifold  ''T''. The non-linear ''σ''-model was introduced by , who named it after a field correspondi ...
* Classical Heisenberg ferromagnet model (spin chain) * Classical Gaudin spin system (Garnier system) * Kaup-Kupershmidt equation * Krichever-Novikov equation * Landau–Lifshitz equation (continuous spin field) *
Benjamin–Ono equation In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. It was introduced by and . The Benjamin–Ono equation is :u_t+uu_x+Hu_=0 where ''H'' i ...
* Degasperis-Procesi equation * Dym equation * Massive Thirring model * Three-wave equation ;Integrable PDEs in 2 + 1 dimensions *
Kadomtsev–Petviashvili equation In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashv ...
*
Davey–Stewartson equation In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by to describe the evolution of a three-dimensional wave-packet on water of finite depth. It is a system of partial differential equations for a complex ( wave-am ...
*
Ishimori equation The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable . Equation The Ishimori equation has the form ...
* Novikov-Veselov equation * The
Belinski–Zakharov transform The Belinski–Zakharov (inverse) transform is a nonlinear transformation that generates new exact solutions of the vacuum Einstein's field equation. It was developed by Vladimir Belinski and Vladimir Zakharov in 1978. The Belinski–Zakharov trans ...
generates a Lax pair for the
Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the for ...
s; general solutions are termed gravitational solitons, of which the
Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
, the Kerr metric and some gravitational wave solutions are examples. ;Exactly solvable statistical lattice models * Ising model in 1- and 2-dimensions * Ice-type model of Lieb * 8-vertex model


See also

* Hitchin system


Related areas

* Mathematical physics *
Soliton In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the me ...
* Painleve transcendents *
Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
* Integrable algorithm


Some key contributors (since 1965)


References

* * * * * * * * * * * * * * * *


Further reading

* * *


External links

*
"SIDE - Symmetries and Integrability of Difference Equations"
a conference devoted to the study of integrable difference equations and related topics.


Notes

{{Integrable systems Dynamical systems Hamiltonian mechanics Partial differential equations