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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Legendre transform | 1 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Legendre transform