Liouville–Arnold theorem

E506853

mathematical theorem theorem in Hamiltonian mechanics

The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.

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All labels observed (5)

Label	Occurrences
Liouville integrability	2
Arnold–Liouville theorem	1
Liouville integrability theorem	1
Liouville theorem on integrable systems	1
Liouville–Arnold theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T5256395 — 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: Liouville–Arnold theorem
Context triple: [Carathéodory–Jacobi–Lie theorem, relatedTo, Liouville–Arnold theorem]

A. Kolmogorov–Arnold–Moser theory
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.
B. 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.
C. 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.
D. 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.
E. 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.
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: Liouville–Arnold theorem
Target entity description: The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.

A. Kolmogorov–Arnold–Moser theory
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.
B. 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.
C. 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.
D. 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.
E. 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.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in Hamiltonian mechanics ⓘ
alsoKnownAs	Liouville integrability theorem ⓘ linked to: Liouville–Arnold theorem Liouville theorem on integrable systems ⓘ linked to: Liouville–Arnold theorem
appliesTo	finite-dimensional Hamiltonian systems ⓘ
assumes	Poisson commutativity of integrals ⓘ compact connected regular level set of integrals ⓘ existence of n independent first integrals in involution ⓘ functional independence of integrals on a level set ⓘ smoothness of integrals ⓘ symplectic manifold phase space ⓘ
category	theorems in classical mechanics ⓘ theorems in symplectic topology ⓘ
concerns	action–angle variables ⓘ completely integrable systems ⓘ integrable Hamiltonian systems ⓘ invariant tori ⓘ quasi-periodic motion ⓘ
describes	local canonical transformation to action–angle variables ⓘ topological structure of invariant sets of integrable systems ⓘ
dimensionCondition	2n-dimensional symplectic manifold with n integrals in involution ⓘ
field	Hamiltonian mechanics ⓘ classical mechanics ⓘ dynamical systems ⓘ symplectic geometry ⓘ
formalSetting	symplectic manifolds and Hamiltonian vector fields ⓘ
guarantees	complete integrability under its hypotheses ⓘ
historicalNote	Arnold formulated the modern geometric version in the 20th century ⓘ Liouville proved an early version in the 19th century ⓘ linked to: Joseph Liouville
implies	Hamiltonian flow is quasi-periodic on invariant tori ⓘ constants of motion become functions of action variables only ⓘ equations of motion become linear in angle variables ⓘ existence of action–angle coordinates near regular invariant tori ⓘ regular common level sets are n-dimensional tori ⓘ
namedAfter	Joseph Liouville ⓘ Vladimir Arnold ⓘ
relatedTo	Kolmogorov–Arnold–Moser theorem ⓘ linked to: Kolmogorov–Arnold–Moser theory Liouville integrability ⓘ linked to: Liouville–Arnold theorem Noether's theorem ⓘ action–angle coordinates ⓘ
requires	Poisson bracket structure ⓘ
usedFor	analysis of planetary motion ⓘ construction of integrable models in classical mechanics ⓘ perturbation theory in celestial mechanics ⓘ study of near-integrable Hamiltonian systems ⓘ

How these facts were elicited