Liouville–Arnold theorem
E506853
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| 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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
Liouville theorem on integrable systems NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Liouville proved an early version in the 19th century NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Vladimir Arnold NERFINISHED ⓘ |
| relatedTo |
Kolmogorov–Arnold–Moser theorem
NERFINISHED
ⓘ
Liouville integrability NERFINISHED ⓘ Noether's theorem NERFINISHED ⓘ 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
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Instruction
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Input
Subject: Liouville–Arnold theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.