In mathematics and physics, there are various distinct notions that are referred to under the name of integrable systems. In the general theory of differential systems, there is Frobenius integrability, which refers to overdetermined systems. In the classical theory of Hamiltonian dynamical systems, there is the notion of Liouville integrability. More generally, in differentiable dynamical systems integrability relates to the existence of foliations by invariant submanifolds within the phase space. Each of these 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This is a discussion of a present category of science. ... In mathematics, a foliation is a geometric device used to study manifolds. ... Phase space of a dynamical system with focal stability. ... In mathematics, a foliation is a geometric device used to study manifolds. ...

Frobenius theorem for overdetermined differential systems

The Frobenius theorem states that an overdetermined differential system is Frobenius integrable; i.e. the space on which it is defined has a foliation by maximal integral manifolds if and only if it generates an ideal that is closed under exterior differentation. (See the article on integrability conditions for differential systems for a detailed discussion of foliations by maximal integral manifolds.) In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ... In mathematics, a foliation is a geometric device used to study manifolds. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... In mathematics, a foliation is a geometric device used to study manifolds. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...

General dynamical systems

In the context of differentiable dynamical systems, integrability refers to an invariant, regular foliation; i.e., one whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the flow. In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ... Immersed submanifold straight line with selfintersections In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. ... In mathematics, flow refers to the group action of a one-parameter group on a set. ...

There is also an extension of the notion of integrability applicable to discrete systems. This definition can be adapted to describe the space of solutions to either a system of differential equations that are evolution equations or difference equations. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

The distinction between integrable and nonintegrable classical 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 explictly integrated in exact form. A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...

Hamiltonian systems and Liouville integrability

In the special setting of Hamiltonian dynamical systems, there is also a distinction between complete integrability, in the Liouville sense, and partial integrability and, a notion of superintegrability and "maximal supertintegrability". Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ...

Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian flows generated by any of the invariants of the foliation preserve the leaves. 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 dimensions, 2n, and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is n. When the number of Poisson commuting invariants is less than n, we say the system is partially integrable. When there exist further functionally independent invariants, besides the maximal number that can be Poisson cummuting, and hence the dimension of the leaves of the invariant folition is less than n, we say the system is superintegrable. If there is a regular foliation with 1-dimensional leaves (curves), this is called maximally superintegrable. In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... Phase space of a dynamical system with focal stability. ... Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms. ...

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 special sets of canonical coordinates on the phase space of the system, known as action-angle variables. The action variables are constants of motion, and the angle variables are the natural linear, periodic coordinates on the torus. The motion on the invariant torus, is linear in the angle variables. In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ... Phase space of a dynamical system with focal stability. ... In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... In geometry, a torus (pl. ...

Quantum integrable systems

There is also 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.

Since there is no clear definition of independence of operators, except for special classes, the definition of integrable system, in the quantum sense, is not yet agreed upon. The working definition that is mostly used is that there is a maximal set of commuting operators, including the Hamiltonian, and a semiclassical limit in which these operators have symbols that are independent Poisson commuting functions on the phase space.

Exactly solvable models and inverse spectral methods

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. In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable is therefore implies solvablility. ...

A resurgence of interest in classical integrable systems came with the discovery, in the late 1960's, that solitons, 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 undersood by viewing these equations as infinite dimensional integrable Hamiltonian systems. Their study lead to a very fruitful approach for "integrating" such systems, the inverse scattering transform and more general inverse spectral methods, which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations. In mathematics and physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed; solitons are caused by a delicate balance between nonlinear and dispersive effects in the medium. ... The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ... In mathematics, the inverse scattering transform is a procedure for integrating certain nonlinear partial differential equations (PDEs) by first converting them into a system of linear ordinary differential equations (ODEs). ...

References

• V.I. Arnold (1997). Mathematical Methods of Classical Mechanics, 2nd ed.. Springer. ISBN 0387968903, ISBN 978-0387968902. 
• L.D. Faddeev, L. A. Takhtajan (1987). Hamiltonian Methods in the Theory of Solitons. Addison-Wesley. ISBN 0387155791, ISBN 978-0387155791. 
• H. Goldstein (1980). Classical Mechanics, 2nd. ed.. Addison-Wesley. ISBN 0-201-02918-9. 
• John P. Harnad, Pavel Winternitz, Gert Sabidussi, eds. (2000). Integrable Systems: From Classical to Quantum. American Mathematical Society. ISBN 0821820931. 
• V.E. Korepin, N. M. Bogoliubov, A. G. Izergin (1997). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. ISBN 9780521586467, ISBN 0521586461.