Hamilton function of a free particle/Landau's book

[fluidistic]

Homework Statement

The problem is taken out of Landau's book on classical mechanics. I must find the Hamilton function and the corresponding Hamilton equations for a free particle in Cartesian, cylindrical and spherical coordinates.

Homework Equations

Hamilton function: $H(p,q,t)= \sum p_i \dot q _i -L$ where L is the Lagrangian.
Hamilton equations: $\dot q_i = \frac{\partial H}{\partial p_i}$ and $\dot p_i =-\frac{\partial H}{\partial q _i}$.

The Attempt at a Solution

I'm stuck on Cartesian coordinates so far. $L=\frac{m}{2}(\dot x ^2 + \dot y^2 +\dot z^2)-U(x,y,z)$.
$p_i=\frac{\partial L}{\partial \dot q _i}=m \dot q_i \Rightarrow \dot p_i=m \ddot q_i$.
$H=m \dot x ^2 +m \dot y^2 +m \dot z ^2 - \frac{m}{2} (\dot x^2 + \dot y^2 +\dot z^2)+U(x,y,z)=\frac{m}{2}(\dot x ^2 +\dot y^2 +\dot z^2)+U(x,y,z)$. I notice that the Hamilton function is the Hamiltonian and that in this case it's worth the total energy of the system (the free particle).
However the solution given in the book is $H=\frac{1}{2m} (p _x ^2+ p_y^2 +p_z ^2)+U(x,y,z)$. Why is it expressed under this form? I took Landau's expressions and definitions and land on a different answer... why?!

Edit: Hmm I guess it's because H should depend only on p, q and t. Never on $\dot q$?

 

[mathfeel]

Lagrangian is a function of $q$, $\dot{q}$, and $t$
Hamiltonian is a function of $q$, $p$, and $t$

You might think that this is just semantics since $\dot{q}$ and $p$ are proportional to each other, but it is very important distinction.

Not sure Landau is the best book to learn these stuff from. It loves elegant solutions and hates verbose explanation.

 

[fluidistic]

Ok thanks a lot for the explanation.
By the way, about the book, do you have any other suggestion? I currently own Goldstein's 1st edition on classical mechanics. I find very few worked examples and lots of theory. So I try to complete with Landau's book.