- #1

RJLiberator

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## Homework Statement

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Find the Legendre Transformation of [tex]f(x)=x^3[/tex]

## Homework Equations

[tex]m(x) = f'(x) = 3x^2[/tex]

[tex] x = {\sqrt{\frac{m(x)}{3}}}[/tex]

[tex]g = f(x)-xm[/tex]

## The Attempt at a Solution

I am reading a quick description of the Legendre Transformation in my required text and it has the example giving for the function [tex]f(x) = \frac{1}{2}e^{2x} [/tex]

Thus, I am trying to follow through with it as follows:

[tex] g=f(x)-xm = x^3-x(3x^2)[/tex]

The problem for me exists in the next step. For them, they had an easy simplification for their example problem. In my problem, m does not equal x^3, m is equal to 3x^2.

Can I do this to solve the transformation:

Since [tex] x = {\sqrt{\frac{m(x)}{3}}}[/tex]

We have

[tex]({\sqrt{\frac{m(x)}{3}}})^3- {\sqrt{\frac{m(x)}{3}}}m = g(m)[/tex]

And that is the Legendre transformation of x^3.

Note* First time ever being exposed to Legendre Transformation.