Legendre transform

The Legendre transform (Adrien-Marie Legendre) is used to perform a change change of variables (see, for example, Ref. 1, Chapter 4 section 11 Eq. 11.20 - 11.25):

If one has the function $f(x,y);$ one can write

$df={\frac {\partial f}{\partial x}}dx+{\frac {\partial f}{\partial y}}dy$

Let $p=\partial f/\partial x$, and $q=\partial f/\partial y$, thus

$df=p~dx+q~dy$

If one subtracts $d(qy)$ from $df$, one has

$df-d(qy)=p~dx+q~dy-q~dy-y~dq$

or

$d(f-qy)=p~dx-y~dq$

Defining the function $g=f-qy$ then

$dg=p~dx+q~dy$

The partial derivatives of $g$ are

${\frac {\partial g}{\partial x}}=p,~~~{\frac {\partial g}{\partial q}}=-y$.

References

1. Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition.