Legendre transformation

E695821

duality transform mathematical transformation

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.

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All labels observed (5)

Label	Occurrences
Legendre transformation canonical	3
Legendre transform	2
Legendre–Fenchel transform	2
Fenchel conjugate	1
Fenchel transform	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]

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
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 ⓘ
property	involutive on suitable convex functions ⓘ order-reversing on convex functions ⓘ preserves convexity via convex conjugation ⓘ
relatedConcept	Fenchel transform ⓘ linked to: Legendre transformation Hamiltonian mechanics ⓘ Lagrangian mechanics ⓘ Legendre–Fenchel transform ⓘ linked to: Legendre transformation 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