Kolmogorov–Arnold–Moser theory

E320432

mathematical theory result in dynamical systems theorem in Hamiltonian dynamics

Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.

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All labels observed (4)

Label	Occurrences
KAM theory	4
Kolmogorov–Arnold–Moser theory canonical	3
Kolmogorov–Arnold–Moser theorem	1
Kolmogorov’s invariant torus theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T3037537 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Kolmogorov–Arnold–Moser theory
Context triple: [Andrei Kolmogorov, notableWork, Kolmogorov–Arnold–Moser theory]

A. Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
B. Kovalevskaya top
The Kovalevskaya top is a famous integrable case of the motion of a rigid body about a fixed point in classical mechanics, discovered and analyzed by mathematician Sofia Kovalevskaya.
C. Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
D. Jacobi integral
The Jacobi integral is a conserved quantity in celestial mechanics and dynamical systems that simplifies the analysis of motion in rotating reference frames, particularly in the restricted three-body problem.
E. Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Kolmogorov–Arnold–Moser theory
Target entity description: Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.

A. Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
B. Kovalevskaya top
The Kovalevskaya top is a famous integrable case of the motion of a rigid body about a fixed point in classical mechanics, discovered and analyzed by mathematician Sofia Kovalevskaya.
C. Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
D. Jacobi integral
The Jacobi integral is a conserved quantity in celestial mechanics and dynamical systems that simplifies the analysis of motion in rotating reference frames, particularly in the restricted three-body problem.
E. Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	mathematical theory ⓘ result in dynamical systems ⓘ theorem in Hamiltonian dynamics ⓘ
abbreviation	KAM theory ⓘ linked to: Kolmogorov–Arnold–Moser theory
appliesTo	area-preserving maps ⓘ nearly integrable Hamiltonian systems ⓘ small perturbations of integrable systems ⓘ symplectic diffeomorphisms ⓘ
assumes	non-degeneracy conditions on the integrable Hamiltonian ⓘ
characterizedBy	control of small divisors via Diophantine conditions ⓘ use of iterative schemes to construct invariant tori ⓘ
describes	persistence of invariant tori ⓘ persistence of quasi-periodic motions ⓘ
explains	stability of quasi-periodic orbits under small perturbations ⓘ survival of invariant tori of positive measure ⓘ
field	Hamiltonian mechanics ⓘ dynamical systems ⓘ mathematical physics ⓘ symplectic geometry ⓘ
hasConsequence	existence of KAM tori in phase space ⓘ long-term stability of planetary motions in celestial mechanics ⓘ
historicalDevelopment	extended by Vladimir Arnold in the 1960s ⓘ initiated by Andrey Kolmogorov in the 1950s ⓘ refined by Jürgen Moser in the 1960s ⓘ
implies	coexistence of regular and chaotic motion ⓘ existence of invariant tori of positive Lebesgue measure ⓘ persistence of most invariant tori under sufficiently small perturbations ⓘ
influenced	development of symplectic topology ⓘ modern theory of Hamiltonian chaos ⓘ
namedAfter	Andrey Kolmogorov ⓘ linked to: Andrei Kolmogorov Jürgen Moser ⓘ Vladimir Arnold ⓘ
relatedTo	Arnold diffusion ⓘ Kolmogorov’s invariant torus theorem ⓘ linked to: Kolmogorov–Arnold–Moser theory Nekhoroshev theory ⓘ standard map ⓘ twist maps ⓘ
studiedIn	accelerator physics ⓘ celestial mechanics ⓘ nonlinear oscillations ⓘ plasma physics ⓘ
usesConcept	Diophantine conditions ⓘ analytic Hamiltonians ⓘ canonical transformations ⓘ non-resonance conditions ⓘ perturbation theory ⓘ small divisor problems ⓘ symplectic maps ⓘ

How these facts were elicited