Legendre transform
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The Legendre transform is used to perform a change of variables (see, for example, Ref.
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
\[dg = p~dx - y~dq\]
The partial derivatives of \(g\) are
\[\frac{\partial g}{\partial x}= p, ~~~ \frac{\partial g}{\partial q}= -y\].
[edit] See also
[edit] References
- ↑ Mary L. Boas "Mathematical methods in the Physical Sciences" John Wiley & Sons, Second Edition ISBN 0471044091
- Related reading
