Nonintegrable

Nonintegrable describes systems that cannot be solved in closed form by quadrature or lack a complete set of first integrals in involution. In mathematics, a system of differential equations is integrable if its solutions can be expressed through a finite sequence of algebraic operations and quadratures; in Hamiltonian mechanics, an autonomous system with n degrees of freedom is integrable if there exist n independent conserved quantities in involution, yielding action-angle variables and solvable motion.

When these conditions fail, the system is nonintegrable. The absence of a full set of integrals typically leads to rich and potentially chaotic dynamics, with trajectories that may diverge from each other and phase space that contains both regular and irregular regions. Nonintegrability is especially discussed in the context of perturbed integrable systems, where small perturbations destroy some invariant structures predicted by the idealized model.

Examples often cited include the general three-body problem in celestial mechanics and the Hénon–Heiles system, both of which exhibit nonintegrable behavior for generic parameter values. The Fermi–Pasta–Ulam–Tsingou problem is another classic case showing departures from simple integrable dynamics, even though near-integrable behavior can occur for certain regimes. The framework of KAM (Kolmogorov–Arnold–Moser) theory describes how some invariant tori survive weak perturbations while others break, creating chaotic regions.

Diagnosing nonintegrability typically relies on numerical indicators such as Lyapunov exponents and Poincaré sections, or on analytical approaches including Ziglin and Morales–Ramis differential Galois theory. Proving nonintegrability is often challenging and results are frequently stated as generic, probabilistic, or conditional.