Hamiltonian as Legendre transformation?

[pellman]

The definition of a Legendre transformation given on the Wikipedia page http://en.wikipedia.org/wiki/Legendre_transformation is: given a function f(x), the Legendre transform f*(p) is

$$f^*(p)=\max_x\left(xp-f(x)\right)$$

Two questions: what does $$\max_x$$ mean here? And why is it not (explicitly?) included in the definition of the Hamiltonian

$$H(q,p)=p\dot{q}-L(q,\dot{q})$$

if the Hamiltonian is a Legendre transformation?

 

[pellman]

I get it. If we want to maximize

$$g(x)=xp-f(x)$$


then we set $$g'(x)=0$$ which is the same as putting $$p=f'(x)$$. In mechanics this amounts to

$$p=\frac{\partial L}{\partial\dot{q}}$$

Well.. thanks to anyone who read and at least thought about replying. :-)