Legendre Transformation

Date:

28 July 2019

Dependency

• Total Derivative

What is the problem ?

Convert a function of one set of variables to another function of their conjugate set of variables (defined below)

Derivation

Step 1 Write a function \(f(x,y)\) in its Total Derivative

\[df=\partial_x f dx + \partial_y f dy \equiv u dx + vdy\]

• \((u,x)\) is a conjugate pair

• \((v,y)\) is a conjugate pair

Step 2 Then one example of Legendre Transformation of \(f(x,y)\) is

\[g \equiv f - xu\]

and \(g\) is a function of \(u,y\), since

\[dg = df - xdu - udx = - xdu + vdy\]

• \(u,y\) are Natural Variables of \(g\)

• \(x,y\) are Natural Variables of \(f\)