I am confused by the Legendre transform. I did the math and carried it out for a function $f(x,y)$, but I don't understand what good it does.
What is the purpose? Didn't I merely just redefine my variables in terms of other variables?
It seems to me just a fancy way of rewriting the function so instead of $x$ and $y$ I now have $u$ and $w$ which are called the conjugate variables.
Can someone explain why the Legendre transform is useful?
"Didn't I merely just redefine my variables in terms of other variables?"
Well, yes. That right there is the point of the Legendre transform. It can be thought of in the same way as integrating by parts (in fact, that's really what you're doing). The only thing that integrating by parts does to an equation is swap around the terms, but often that simplifies the integral. In the same way, the Legendre transform can simplify some equations.