Legendre Transformation
=================================

:Date: 28 July 2019

Dependency
-------------

- `Total Derivative`_

.. _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 :math:`f(x,y)` in its `Total Derivative`_

.. math::
	df=\partial_x f dx + \partial_y f dy \equiv u dx + vdy

- :math:`(u,x)` is a conjugate pair
- :math:`(v,y)` is a conjugate pair	

**Step 2** Then **one example of** Legendre Transformation of :math:`f(x,y)` is

.. math::
	g \equiv f - xu

and :math:`g` is a function of :math:`u,y`, since

.. math::
	dg = df - xdu - udx = - xdu + vdy 

- :math:`u,y` are **Natural Variables** of :math:`g`
- :math:`x,y` are **Natural Variables** of :math:`f`