Carathéodory–Jacobi–Lie theorem
E118708
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Carathéodory–Jacobi–Lie theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T998596 — 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: Carathéodory–Jacobi–Lie theorem Context triple: [Constantin Carathéodory, notableWork, Carathéodory–Jacobi–Lie theorem]
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A.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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C.
Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
-
D.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
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E.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
- 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: Carathéodory–Jacobi–Lie theorem Target entity description: 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.
-
A.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
C.
Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
-
D.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
-
E.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
- F. None of above. chosen
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in Hamiltonian mechanics ⓘ theorem in symplectic geometry ⓘ |
| appliesTo |
Poisson manifolds
ⓘ
symplectic manifolds ⓘ |
| assumes |
a set of pairwise Poisson-commuting functions
ⓘ
functional independence of the given functions on an open set ⓘ |
| category |
theorem in geometry
ⓘ
theorem in mathematical analysis ⓘ |
| concerns |
Poisson-commuting integrals of motion
ⓘ
normal forms of Hamiltonian systems near regular points ⓘ |
| concludes |
existence of local canonical coordinates
ⓘ
the given commuting functions depend only on a subset of the canonical coordinates ⓘ |
| context |
canonical coordinate systems in mechanics
ⓘ
local structure of symplectic manifolds ⓘ |
| field |
Hamiltonian mechanics
ⓘ
differential geometry ⓘ mathematical physics ⓘ symplectic geometry ⓘ |
| generalizes | Darboux theorem ⓘ |
| guarantees |
adaptation of canonical coordinates to a given integrable family of functions
ⓘ
existence of coordinates in which the symplectic form has standard canonical form ⓘ |
| namedAfter |
Carl Gustav Jacob Jacobi
ⓘ
Constantin Carathéodory ⓘ Sophus Lie ⓘ |
| provides | canonical local coordinates adapted to a given set of commuting functions ⓘ |
| relatedTo |
Darboux theorem
ⓘ
Liouville–Arnold theorem ⓘ Poisson brackets ⓘ symplectic form ⓘ |
| subjectOf |
Darboux-type coordinate systems
ⓘ
action–angle variables ⓘ canonical local coordinates ⓘ commuting functions ⓘ integrable Hamiltonian systems ⓘ |
| usedIn |
Hamiltonian mechanics
ⓘ
canonical transformations ⓘ classical mechanics ⓘ theory of integrable systems ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Carathéodory–Jacobi–Lie theorem Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.