# What is Legendre transformation: Definition and 18 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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1. ### A Expression of Shot noise when expanding ##a_{\ell m}## coefficients

I would like to arrive at the following expression for the quantity ##o_{\ell}## ( with "DM" for Dark Matter ): ##o_{\ell}=b_{s p}^2 C_{\ell}^{D M}+B_{s p}## with Poisson noise ##B_{s p}=\frac{1}{\bar{n}}(\bar{n}## being the average number of galaxies observed). the index "sp" is for spectro...
2. ### Apply the Legendre Transformation to the Entropy S as a function of E

Hi, Unfortunately I am not getting anywhere with task three, I don't know exactly what to show Shall I now show that from ##S(T,V,N)## using Legendre I then get ##S(E,V,N)## and thus obtain the Sackur-Tetrode equation?
3. ### I Legendre Transformation & Hamilton-Jacobi Formalism: A Relationship?

Hey I have a question about the relation between Legendre transformation and Hamilton-Jacobi formalism. Is there some relation? Cause Hamilton-Jacobi is the expression of Hamiltonian with a transformation.
4. ### I Legendre Transform: 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} ##...
5. ### Hamiltonian mechanics - the independence of p and q

in the Lagrangian mechanics, we assumed that the Lagrangian is a function of space coordinates, time and the derivative of those space coordinates by time (velocity) L(q,dq/dt,t). to derive the Hamiltonian we used the Legendre transformation on L with respect to dq/dt and got H = p*(dq/dt) -...
6. ### Discrepancy in Lagrangian to Hamiltonian transformation?

I know, $$L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian)$$ $$H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads to the equation,  H - L= 2V \;\;\; \; \;\;\...
7. ### A Chain rule - legendre transformation

let df=∂f/∂x dx+∂f/∂y dy and ∂f/∂x=p,∂f/∂y=q So we get df=p dx+q dy d(f−qy)=p dx−y dqand now, define g. g=f−q y dg = p dx - y dq and then I faced problem. ∂g/∂x=p←←←←←←←←←←←←←←← book said like this because we can see g is a function of x and p so that chain rule makes ∂g/∂x=p but I wrote...
8. ### 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}...
9. ### Legendre Transformation of f(x) = x^3

Homework Statement [/B] Find the Legendre Transformation of f(x)=x^3 Homework Equations m(x) = f'(x) = 3x^2 x = {\sqrt{\frac{m(x)}{3}}} g = f(x)-xm The Attempt at a Solution I am reading a quick description of the Legendre Transformation in my required text and it has the example giving for...
10. ### 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...
11. ### 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...
12. ### Helmholtz Free Energy Legendre Transformation

Homework Statement Show how a Legendre transformation is used to obtain the Helmholtz free energy A(T,V) from the internal energy and derive the general expression for the differential of A. Homework Equations Internal Energy is a function of Entropy and Volume. U Ξ (S, V) A Ξ (T,V) A = U...
13. ### Legendre Transformation of Lagrangian density ?

Hi I began to study the basics of QED. Now I am studying Lagrangian and Hamiltonian densities of Dirac Equation. I'll call them L density and H density for convenience :)Anyway, the derivation of the H density from L density using Legendre transformation confuses me :( I thought because...
14. ### Thermodynamics, Helmholtz free energy, Legendre transformation

Homework Statement The Helmholtz free energy of a certain system is given by F(T,V) = -\frac{VT^2}{3}. Calculate the energy U(S,V) with a Legendre transformation. Homework Equations F = U - TS S = -\left(\frac{\partial F}{\partial T}\right)_V The Attempt at a Solution We...
15. ### Legendre Transformation of the Hamiltonian

It's given as this H\left(q_i,p_j,t\right) = \sum_m \dot{q}_m p_m - L(q_i,\dot q_j(q_h, p_k),t) \,. But if it's a Legendre transformation, then couldn't you also do this? H\left(q_i,p_j,t\right) = \sum_m \dot{p}_m q_m - L(p_i,\dot p_j(p_h, q_k),t) \,.
16. ### Legendre transformation of the CR3BP equations

I'm not quite sure where to post this but I suppose it should go here given it's about classical mechanics... Anyhoo. I'm currently on the long road to implementing a symplectic integrator to simulate the closed restricted 3 body problem and I'm in the process of deriving the Hamiltonian...
17. ### Hamiltonian as Legendre transformation?

The definition of a Legendre transformation given on the Wikipedia page http://en.wikipedia.org/wiki/Legendre_transformation is: given a function f(x), the Legendre transform f*(p) is f^*(p)=\max_x\left(xp-f(x)\right) Two questions: what does \max_x mean here? And why is it not...
18. ### How Does the Legendre Transformation Apply to Thermodynamics with Tension Force?

Homework Statement At an elastic bar we give work because of hydrostatic pressure P and applied tension force F at the axis length that has length l. Homework Equations 1) Give an expression for dU. 2) With the help of legendre transformations find the thermodynamic equations and the...