nonintegrability

Nonintegrability is a property of a dynamical system in which there are not enough independent conserved quantities to allow the equations of motion to be solved by quadrature. In Hamiltonian mechanics, a system with n degrees of freedom is Liouville integrable if there exist n independent first integrals that are in involution with respect to the Poisson bracket; when such a full set of integrals is absent, the system is nonintegrable. Nonintegrability does not imply randomness in every trajectory, but it generally prevents a complete analytic solution and leads to more complex dynamics than in integrable cases.

In integrable systems, motion is highly regular and often confined to invariant tori. By contrast, nonintegrable systems can exhibit a mixture of regular and chaotic behavior, with trajectories that diverge rapidly from nearby initial conditions. The transition from order to chaos is a central theme in the study of nonlinear dynamics. In near-integrable systems, Kolmogorov-Arnold-Moser (KAM) theory shows that many invariant tori persist under small perturbations, but resonant tori break and chaotic regions emerge.

Common examples of nonintegrable systems include the Henon–Heiles system, the general three-body problem, and the double pendulum. These systems are typically solvable only through numerical methods or qualitative analyses rather than exact closed-form solutions.

Assessment of nonintegrability often relies on numerical and qualitative tools such as Poincaré sections, Lyapunov exponents, and frequency analysis. Nonintegrability is a property of the system as a whole, while individual trajectories may still display regular motion within chaotic regions.