In mathematics,integrability is a property of certaindynamical systems. While there are several distinct formal definitions, informally speaking, anintegrable system is a dynamical system with sufficiently manyconserved quantities, orfirst integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of itsphase space.
Three features are often referred to as characterizing integrable systems:[1]
Integrable systems may be seen as very different in qualitative character from moregeneric dynamical systems,which are more typicallychaotic systems. 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 theHamiltonian 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 (theEuler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (theLagrange top).
In the late 1960s, it was realized that there arecompletely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (Korteweg–de Vries equation), theKerr effect in optical fibres, described by thenonlinear Schrödinger equation, and certain integrable many-body systems, such as theToda lattice. The modern theory of integrable systems was revived with the numerical discovery ofsolitons byMartin Kruskal andNorman Zabusky in 1965, which led to theinverse scattering transform method in 1967.
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 (theleaves of theLagrangian foliation), and if the flows are complete and the energy level set is compact, this implies theLiouville–Arnold theorem; i.e., the existence ofaction-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomousHamiltonian 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 theFrobenius theorem, which states that a system isFrobenius integrable (i.e., is generated by an integrable distribution) if, locally, it has afoliation by maximal integral manifolds. But integrability, in the sense ofdynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.
Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set ofspecial functions; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.
In the context of differentiabledynamical systems, the notion ofintegrability refers to the existence of invariant, regularfoliations; i.e., ones whose leaves areembedded submanifolds of the smallest possible dimension that are invariant under theflow. 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 ofHamiltonian systems, known ascomplete integrability in the sense ofLiouville (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 ofdifferential equations or finite difference equations.
The distinction between integrable and nonintegrable dynamical systems 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 an exact form.
In the special setting ofHamiltonian systems, we have the notion of integrability in theLiouville sense. (See theLiouville–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 functionally independentPoisson commuting invariants (i.e., independent functions on the phase space whosePoisson brackets with the Hamiltonian of the system, and with each other, vanish).
In finite dimensions, if thephase space issymplectic (i.e., the center of the Poisson algebra consists only of constants), it must have even dimension and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is. The leaves of the foliation aretotally isotropic with respect to the symplectic form and such a maximal isotropic foliation is calledLagrangian. Allautonomous 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 aretori, 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 calledaction-angle variables (see below).
There is also a distinction betweencomplete integrability, in theLiouville sense, and partial integrability, as well as a notion ofsuperintegrability 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 issuperintegrable. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable.
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 aretori. There then exist, as mentioned above, special sets ofcanonical coordinates on thephase space known asaction-angle variables,such that the invariant tori are the joint level sets of theaction 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 tori. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.
Incanonical transformation theory, there is theHamilton–Jacobi method, in which solutions to Hamilton's equations are sought by first finding a complete solution of the associatedHamilton–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 theaction-angle variables. In the general theory of partial differential equations ofHamilton–Jacobi type, a complete solution (i.e. one that depends onn independent constants of integration, wheren 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 theHamilton–Jacobi equation is by no means a characterization of complete integrability in the Liouville sense. Most cases that can be "explicitly integrated" involve a completeseparation 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.
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, thatsolitons, which are strongly stable, localized solutions of partial differential equations like theKorteweg–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, theinverse scattering transform and more general inverse spectral methods (often reducible toRiemann–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 tocompletely 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.
Another viewpoint that arose in the modern theory of integrable systems originated ina calculational approach pioneered byRyogo Hirota,[2] which involved replacingthe original nonlinear dynamical system with a bilinear system of constant coefficientequations for an auxiliary quantity, which later came to be known as theτ-function. These are now referred to as theHirota equations. Although originally appearing just as a calculational device, without any clear relationto theinverse scattering approach, or the Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such assolitons could be derived.
Subsequently, this was interpreted byMikio Sato[3] and his students,[4][5] at first for the case of integrable hierarchies of PDEs, such as theKadomtsev–Petviashvili hierarchy, but then for much more general classes of integrable hierarchies, as a sort ofuniversal phase space approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abeliangroup action on a (finite or infinite)Grassmann manifold. The τ-function was viewed as thedeterminantof aprojection operator from elements of thegroup orbit to someorigin within the Grassmannian,and theHirota equations as expressing thePlücker relations, characterizing the Plücker embedding of the Grassmannian in the projectivization of a suitably defined (infinite)exterior space, viewed as afermionic Fock space.
There is also a notion of quantum integrable systems.
In the quantum setting, functions on phase space must be replaced byself-adjoint operators on aHilbert space, and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized tolocal conservation laws.[6] EveryHamiltonian has an infinite set of conserved quantities given by projectors to its energyeigenstates. 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. TheYang–Baxter 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 thequantum inverse scattering method where the algebraicBethe ansatz can be used to obtain explicit solutions. Examples of quantum integrable models are theLieb–Liniger model, theHubbard model and several variations on theHeisenberg model.[7] Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model.[8]
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: theBethe ansatz approach, in its modern sense, based on theYang–Baxter equations and thequantum 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]