In mathematics and physics, an integrable system refers to a system of partial differential equations that may be integrated to obtain the solutions to the equations. In physics, the distinction between integrable and non-integrable systems is usually a distinction between systems with regular motion and chaotic motion. In the mathematical study of differential equations, the Frobenius theorem is a major result: it states that a system is integrable only if it has a foliation. The article on integrability conditions for differential systems discusses the general case in mathematical terms.

A particularly vital area of modern mathematical research is a class of integrable systems known as exactly solvable models. These tend to be sets of non-linear differential equations which have non-intuitive constants of motion, solutions that are soliton, and a rich mathematical theory.

Hamiltonian mechanics

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In Hamiltonian mechanics, an integrable system refers to a Hamiltonian system that has constants of motion other than the energy itself. A completely integrable system is a system that has n degrees of freedom, n constants of motion, and whose constants of motion are in involution: that is, the Poisson bracket betweenss each pair of constants of motion vanishes.

When a system is completely integrable, there is a special set of variable on the phase space of the system, known as action-angle coordinates. The actions are the constants of motion, and all motion occurs on the surface of a torus, known as the invariant torus. The coordinates on the torus are the angle variables.

Systems which are not completely integrable are in general chaotic.

General considerations

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There are a number of topics and theorems that deal with integrability. These include Liouville's theorem, which describes the behaviour of the measure on phase space as a function of time. Noether's theorem states that for every constant of motion, there is a corresponding continuous symmetry of the system under consideration. The Frobenius theorem states that a set of differential equations can be integrated only when the manifold which they describe has a foliation. The article on integrability conditions for differential systems discusses the general case in mathematical terms.

Dynamical systems | Hamiltonian mechanics | Partial differential equations