integrables

Integrables, in mathematics and physics, refers to systems or equations that can be solved exactly due to a high degree of structure, typically a large number of conserved quantities. The term covers finite-dimensional classical systems, infinite-dimensional systems described by partial differential equations, and quantum or discrete models. Central to many notions of integrability is the idea that the equations of motion can be integrated by analytic methods, often leading to explicit formulas for the evolution of the system.

In classical mechanics, a standard notion is Liouville integrability. A Hamiltonian system with 2n-dimensional phase space is Liouville integrable if it possesses n independent integrals of motion that are in involution (their Poisson brackets vanish pairwise). Under suitable regularity conditions, the motion can be described by action-angle variables, and the equations are solvable by quadrature. Examples include the harmonic oscillator and the Kepler problem. By contrast, generic many-body systems, such as the general n-body problem, are not integrable.

For many integrable systems, powerful algebraic and analytic methods exist. A common framework uses Lax pairs and the associated spectral parameter, leading to an infinite hierarchy of conserved quantities and, for nonlinear PDEs, to inverse scattering techniques and soliton solutions. Well-known infinite-dimensional integrable PDEs include the Korteweg–de Vries and nonlinear Schrödinger equations.

Quantum integrable systems feature a commuting family of operators, enabling exact spectra via methods such as the Bethe ansatz. Examples include the Heisenberg spin chain and the Lieb–Liniger model. Discrete and lattice integrable systems similarly preserve an abundance of conserved quantities under evolution.

Integrability is a unifying theme across algebra, geometry, and analysis, distinguishing exactly solvable models by their rich, rigid structure.