Legendre transformation
E695821
The Legendre transformation is a mathematical operation that converts a function of one set of variables into a function of their conjugate variables, widely used in classical mechanics and thermodynamics to switch between different energy or potential formulations.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Legendre transform | 1 |
| Legendre transformation canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7861129 — 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: Legendre transformation Context triple: [Adrien-Marie Legendre, knownFor, Legendre transformation]
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A.
Landen transformations
Landen transformations are classical iterative formulas in analysis that relate elliptic integrals (and associated means) at different moduli, enabling their efficient evaluation and simplification.
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B.
Bogoliubov transformation
The Bogoliubov transformation is a mathematical change of basis used in quantum field theory and many-body physics to diagonalize Hamiltonians by mixing creation and annihilation operators, enabling the description of quasiparticles and phenomena like superconductivity.
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C.
Laplace transform
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
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D.
Lagrangian function
The Lagrangian function is a mathematical construct that combines an objective function with its constraints, widely used in optimization and variational calculus to analyze and solve constrained problems.
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E.
Sommerfeld-Watson transform
The Sommerfeld-Watson transform is a complex-analysis technique that converts discrete sums over angular momentum into contour integrals, widely used in scattering theory and Regge theory to study analytic properties of amplitudes.
- 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: Legendre transformation Target entity description: The Legendre transformation is a mathematical operation that converts a function of one set of variables into a function of their conjugate variables, widely used in classical mechanics and thermodynamics to switch between different energy or potential formulations.
-
A.
Landen transformations
Landen transformations are classical iterative formulas in analysis that relate elliptic integrals (and associated means) at different moduli, enabling their efficient evaluation and simplification.
-
B.
Bogoliubov transformation
The Bogoliubov transformation is a mathematical change of basis used in quantum field theory and many-body physics to diagonalize Hamiltonians by mixing creation and annihilation operators, enabling the description of quasiparticles and phenomena like superconductivity.
-
C.
Laplace transform
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
-
D.
Lagrangian function
The Lagrangian function is a mathematical construct that combines an objective function with its constraints, widely used in optimization and variational calculus to analyze and solve constrained problems.
-
E.
Sommerfeld-Watson transform
The Sommerfeld-Watson transform is a complex-analysis technique that converts discrete sums over angular momentum into contour integrals, widely used in scattering theory and Regge theory to study analytic properties of amplitudes.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
duality transform
ⓘ
mathematical transformation ⓘ |
| appliedIn |
economics
ⓘ
information theory ⓘ large deviation theory ⓘ statistical mechanics ⓘ variational calculus ⓘ |
| assumes |
duality between extensive and intensive thermodynamic variables
ⓘ
duality between position-like and momentum-like variables ⓘ |
| coreDefinition | given f(x) its transform is g(p)=sup_x(px−f(x)) ⓘ |
| domain |
convex functions
ⓘ
real-valued functions ⓘ |
| field |
classical mechanics
ⓘ
convex analysis ⓘ mathematics ⓘ optimization theory ⓘ thermodynamics ⓘ |
| generalizationOf | simple change of variables between x and its gradient ⓘ |
| hasNotation | f* for the Legendre–Fenchel transform of f ⓘ |
| historicalPeriod | 19th century mathematics ⓘ |
| inverseOf | itself on appropriate function classes ⓘ |
| mapsFrom | function of a variable x ⓘ |
| mapsTo | function of conjugate variable p ⓘ |
| namedAfter | Adrien-Marie Legendre NERFINISHED ⓘ |
| property |
involutive on suitable convex functions
ⓘ
order-reversing on convex functions ⓘ preserves convexity via convex conjugation ⓘ |
| relatedConcept |
Fenchel transform
NERFINISHED
ⓘ
Hamiltonian mechanics NERFINISHED ⓘ Lagrangian mechanics NERFINISHED ⓘ Legendre–Fenchel transform NERFINISHED ⓘ convex conjugate ⓘ thermodynamic potentials ⓘ |
| requires |
convexity for global invertibility
ⓘ
differentiability for classical formula p = df/dx ⓘ |
| usedFor |
changing variables from primal to conjugate variables
ⓘ
constructing Hamiltonian from Lagrangian ⓘ constructing Lagrangian from Hamiltonian when possible ⓘ defining convex conjugates ⓘ deriving Maxwell relations in thermodynamics ⓘ deriving equations of motion in Hamiltonian mechanics ⓘ duality in convex optimization ⓘ passing from internal energy to Gibbs free energy ⓘ passing from internal energy to Helmholtz free energy ⓘ passing from internal energy to enthalpy ⓘ re-expressing variational problems in dual variables ⓘ switching between different energy formulations ⓘ switching between different thermodynamic potentials ⓘ |
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: Legendre transformation Description of subject: The Legendre transformation is a mathematical operation that converts a function of one set of variables into a function of their conjugate variables, widely used in classical mechanics and thermodynamics to switch between different energy or potential formulations.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Legendre transform