Legendre transform

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The Legendre transform is used to perform a 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


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

Defining the function g=f-qy then

dg =  p~dx - y~dq

The partial derivatives of g are

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

See also[edit]


  1. Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition ISBN 0471044091
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