Integrability

Integrability is a property of a mathematical model indicating that its equations of motion can be solved exactly, often by explicit formulas or by quadrature. In classical mechanics, a Hamiltonian system with n degrees of freedom is Liouville integrable if there exist n independent constants of motion in involution (their Poisson brackets vanish pairwise) on a region of phase space. When these integrals exist, the system can be reduced to action-angle variables, and its trajectories lie on invariant n-tori. Such systems are highly regular and typically nonchaotic; their motion can be described by simple, quasi-periodic motion.

Integrability is fragile under perturbations. Kolmogorov–Arnold–Moser theory shows that many invariant tori persist for small perturbations, leading to a mixed phase space with both regular and chaotic regions.

In quantum mechanics, integrability means the existence of a complete set of commuting conserved quantities, allowing the spectrum and eigenstates to be determined exactly for certain models. Quantum integrable systems include those solvable by Bethe ansatz or algebraic methods; their spectral statistics often differ from chaotic systems.

For nonlinear partial differential equations, integrability refers to infinite-dimensional systems possessing an infinite sequence of conserved quantities and solvable by methods such as the inverse scattering transform and Lax pairs. Prototypical examples are the Korteweg–de Vries equation, the nonlinear Schrödinger equation, and the sine-Gordon equation; they admit soliton solutions.

Common examples of integrable finite-dimensional systems are the harmonic oscillator and the Kepler problem; the Toda lattice is a famous many-body integrable system.

Integrability provides exact benchmarks, guides qualitative understanding of dynamics, and helps illuminate the transition to chaos and complexity.