Integrability

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; 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 (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. 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 sufficiently large time.
Complete integrability is thus a nongeneric property of dynamical systems. Nevertheless, many systems studied in physics are completely integrable, in particular, in the 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) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top).
The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform 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, 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. General dynamical systems have no such conserved quantities; in the case of autonomous 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 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.

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