Kolmogorov–Arnold–Moser theorem

The Kolmogorov–Arnold–Moser theorem states that many invariant tori in nearly integrable Hamiltonian systems persist under sufficiently small perturbations when suitable nondegeneracy and frequency conditions hold.

Invariant torus: An invariant torus is a torus-shaped subset of phase space preserved by a dynamical system's evolution. These are the geometric structures whose persistence the theorem guarantees.
Andrey Kolmogorov: Andrey Kolmogorov was a Soviet mathematician whose work helped found modern probability theory and shaped dynamical systems. His 1954 announcement supplied the theorem's first proof outline.
Integrable system: An integrable system has enough independent conserved quantities for its motion to be solved by quadratures. The theorem starts from integrable dynamics, where phase space is organized by invariant tori.
Celestial mechanics: Celestial mechanics applies mathematical models to the motion of planets, moons, asteroids, and spacecraft. The theorem addresses whether regular orbital motions survive weak gravitational interactions.
Arnold diffusion: Arnold diffusion is slow transport through action space in certain nearly integrable Hamiltonian systems with at least three degrees of freedom. It shows that surviving KAM tori do not necessarily prevent all long-term drift.
Hamiltonian system: A Hamiltonian system is a dynamical system whose evolution is generated by a Hamiltonian function. The theorem concerns perturbations of integrable systems in this broader dynamical framework.
Vladimir Arnold: Vladimir Arnold was a Soviet and Russian mathematician known for work in dynamical systems, geometry, and classical mechanics. His 1963 work established a major version for nearly integrable Hamiltonian systems.
Action-angle coordinates: Action-angle coordinates express integrable Hamiltonian motion through conserved actions and evolving angular variables. They put the unperturbed motion into the form used to analyze torus persistence.
Solar System stability: Solar System stability concerns the long-term behavior and boundedness of planetary orbits. KAM provided a mathematical route to stability results for nearly integrable planetary models.
Hamiltonian chaos: Hamiltonian chaos is irregular, sensitive dynamics in systems governed by Hamiltonian mechanics. The theorem helps explain how regular invariant regions can coexist with chaotic motion.

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TorusRelated: It explains when toroidal quasiperiodic motion survives perturbation.