Jacobi bracket
E182756
Lie bracket on functions
bilinear operation
bracket operation
generalization of Poisson bracket
structure in differential geometry
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobi bracket canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1615225 — 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: Jacobi bracket Context triple: [Carl Gustav Jacob Jacobi, notableWork, Jacobi bracket]
-
A.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
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B.
Peierls bracket
The Peierls bracket is a covariant generalization of the Poisson bracket used in quantum field theory and classical field theory to define commutation relations in a way that respects spacetime causality.
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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.
Lie derivative
The Lie derivative is a fundamental differential operator in differential geometry that measures how a tensor field changes along the flow generated by a vector field.
-
E.
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.
- 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: Jacobi bracket Target entity description: The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
-
A.
Lie bracket
The Lie bracket is a bilinear, antisymmetric operation on a Lie algebra that measures the noncommutativity of its elements and encodes its infinitesimal structure.
-
B.
Peierls bracket
The Peierls bracket is a covariant generalization of the Poisson bracket used in quantum field theory and classical field theory to define commutation relations in a way that respects spacetime causality.
-
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.
Lie derivative
The Lie derivative is a fundamental differential operator in differential geometry that measures how a tensor field changes along the flow generated by a vector field.
-
E.
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.
- F. None of above. chosen
Statements (34)
| Predicate | Object |
|---|---|
| instanceOf |
Lie bracket on functions
ⓘ
bilinear operation ⓘ bracket operation ⓘ generalization of Poisson bracket ⓘ structure in differential geometry ⓘ |
| appearsIn |
Hamiltonian formulation on contact manifolds
ⓘ
generalized Hamiltonian dynamics ⓘ |
| associatedWith | local Lie algebra of smooth functions ⓘ |
| characterizes | Jacobi structure on a manifold ⓘ |
| definedOn | space of smooth functions on a Jacobi manifold ⓘ |
| field |
Jacobi geometry
ⓘ
Poisson geometry ⓘ differential geometry ⓘ mathematical physics ⓘ symplectic geometry ⓘ |
| generalizes | Poisson bracket ⓘ |
| hasRole |
defines Hamiltonian vector fields on Jacobi manifolds
ⓘ
encodes infinitesimal symmetries on Jacobi manifolds ⓘ |
| mathematicalDomain |
differential operators
ⓘ
smooth manifolds ⓘ |
| property |
bilinear over the real numbers
ⓘ
first-order differential operator in each argument ⓘ local in nature ⓘ satisfies Jacobi identity ⓘ skew-symmetric ⓘ |
| reducesTo | Poisson bracket when the Jacobi structure is exact ⓘ |
| relatedTo |
Jacobi manifold
ⓘ
Poisson geometry ⓘ
surface form:
Poisson manifold
contact manifold ⓘ |
| satisfies | Leibniz-type rule with respect to pointwise product of functions ⓘ |
| usedIn |
Hamiltonian systems
ⓘ
Jacobi manifold theory ⓘ contact geometry ⓘ local Lie algebra structures ⓘ |
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: Jacobi bracket Description of subject: The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Carl Gustav Jacob Jacobi