# Legendre transformation Definition and 7 Discussions

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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2. ### I Legendre transform and momentum/velocity

I apologize for the simplicity of the question. I have been reading a paper on the Legendre transform (https://arxiv.org/pdf/0806.1147.pdf), and I am not understanding a particular step in the discussion. In the paper, Equation 16, where ##\mathcal{H} = \sqrt{\vec{p}^2 + m^2} ##...
3. ### Hamiltonian mechanics - the independence of p and q

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4. ### Discrepancy in Lagrangian to Hamiltonian transformation?

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5. ### A Conjugate variables in the Fourier and Legendre transforms

In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...
6. ### How do I set up this Legendre Transform for Hamiltonian

Homework Statement Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...
7. ### A Variation of Lagrangian w/r to canonical momenta

Hi, I've been working through Cornelius Lanczos book "The Variational Principles of Mechanics" and there's something I'm having difficulty understanding on page 168 of the Dover edition (which is attached). After introducing the Legendre transformation and transforming the Lagrangian equations...