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. :-)