Legendre transformation

In mathematics and physics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.
For sufficiently smooth functions on the real line, the Legendre transform




f

∗




{\displaystyle f^{*}}
of a function



f


{\displaystyle f}
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 Euler's derivative notation as




D
f
(
⋅
)
=


(

D

f

∗



)


−
1


(
⋅
)

,


{\displaystyle Df(\cdot )=\left(Df^{*}\right)^{-1}(\cdot )~,}
where



(
ϕ

)

−
1


(
⋅
)


{\displaystyle (\phi )^{-1}(\cdot )}
means a function such that



(
ϕ

)

−
1


(
ϕ
(
x
)
)
=
x

,


{\displaystyle (\phi )^{-1}(\phi (x))=x~,}
or, equivalently, as




f
′

(

f

∗
′


(

x

∗


)
)
=

x

∗




{\displaystyle f'(f^{*\prime }(x^{*}))=x^{*}}
and




f

∗
′


(

f
′

(
x
)
)
=
x


{\displaystyle f^{*\prime }(f'(x))=x}
in 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.

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