Definitions
The definition of the Legendre transform can be made more explicit. To extremise with respect to, we set its derivative equal to zero:
Thus, the expression is extremised when
When is convex, this is a maximum because the second derivative of is negative:
Next we invert (2) to obtain as a function of and plug this into (1), which gives the more useful form,
This definition gives the conventional procedure for calculating the Legendre transform of : find, solve for in terms of and substitute into the expression . This definition makes clear the following interpretation: the Legendre transform produces a new function, in which the independent variable is replaced by, which is the derivative of the original function with respect to .
Read more about this topic: Legendre Transformation
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