Editing Legendre transformation (section)

{{Short description|Mathematical transformation}}
{{about|an involution transform commonly used in classical mechanics and thermodynamics|the integral transform using Legendre polynomials as kernels|Legendre transform (integral transform)}}
[[Image:Legendre transformation.png|thumb|256px|right|The function <math>f(x)</math> is defined on the interval <math display="inline">[a,b]</math>. For a given <math>p</math>, the difference <math>px - f(x)</math> takes the maximum at <math>x'</math>. Thus, the Legendre transformation of <math>f(x)</math> is <math>f^*(p) =p x'-f(x')</math>.]]
In [[mathematics]], the '''Legendre transformation''' (or '''Legendre transform'''), first introduced by [[Adrien-Marie Legendre]] in 1787 when studying the minimal surface problem,<ref name=":0">{{Cite book |last=Legendre |first=Adrien-Marie |url=https://www.biodiversitylibrary.org/page/28011033 |title=Mémoire sur l'intégration de quelques équations aux différences partielles. In Histoire de l'Académie royale des sciences, avec les mémoires de mathématique et de physique |publisher=Imprimerie royale |year=1789 |volume= 1787|location=Paris |pages=309–351 |language=French}}</ref> is an [[involution (mathematics)|involutive]] [[List of transforms|transformation]] on [[real number|real]]-valued functions that are [[Convex function|convex]] on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function. 

In physical problems, the Legendre transform is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the [[Conjugate variables (thermodynamics)|conjugate quantity]] (momentum, volume, and entropy, respectively). In this way, it is commonly used in [[classical mechanics]] to derive the [[Hamiltonian mechanics|Hamiltonian]] formalism out of the [[Lagrangian mechanics|Lagrangian]] formalism (or vice versa) and in [[thermodynamics]] to derive the [[Thermodynamic potential|thermodynamic potentials]], as well as in the solution of [[Differential equation|differential equations]] of several variables. 

For sufficiently smooth functions on the real line, the Legendre transform <math>f^*</math> of a function <math>f</math> can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in [[Notation for differentiation#Euler.27s notation|Euler's derivative notation]] as
<math display="block">Df(\cdot) = \left( D f^* \right)^{-1}(\cdot)~,</math> where <math>D</math> is an operator of differentiation, <math>\cdot</math> represents an argument or input to the associated function, <math>(\phi)^{-1}(\cdot)</math> is an inverse function such that <math>(\phi) ^{-1}(\phi(x))=x</math>, or equivalently, as <math>f'(f^{*\prime}(x^*)) = x^*</math> and <math>f^{*\prime}(f'(x)) = x</math> in [[Notation for differentiation#Lagrange's notation|Lagrange's notation]].

The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the [[convex conjugate]] (also called the Legendre–Fenchel transformation), which can be used to construct a function's [[Convex_hull#Functions|convex hull]].