What makes phase space special?

[thegreenlaser]

In Lagrangian/Hamiltonian mechanics, what is it that makes phase space special compared to configuration space? As a simple example, if I use $q$ as my generalized position and $v = \dot{q}$ as my generalized momentum, then the Hamiltonian
$$H = \frac{1}{2} v^2 + \frac{1}{m} V(q)$$
gives the correct dynamics. Yet we always seem make a big deal out of transitioning from configuration space $(q, \dot{q} )$ to phase space $(q, m \dot{q} )$. Is there something about the phase space coordinates $(q, \dot{q} )$ that doesn't work? (From a mathematical point of view, is it maybe impossible to set up a symplectic structure with those coordinates?)

To put it another way, if I arbitrarily pick variables to be my generalized momentum and position and I can find a Hamiltonian which gives the correct dynamics in terms of those variables, is that sufficient to guarantee I've set up a phase space correctly (symplectic structure and all)?

 

[ShayanJ]

There are two points I should say in this respect:
1) Configuration space is not something like phase space with the only difference of having $\dot q$s instead of $p$s. Consider N particles moving in three dimensional space. Their configuration space is a 3N dimensional space where each dimension is either x,y or z of one of the particles. But phase space is the 6N dimensional space where you add the components of the momenta of the particles as the dimensions of the space too.
2) The point of having a 6N dimensional space instead of a 3N dimensional space is that in the former, each point of the space is a unique state of the system while you may have distinct states being in the same point in a 3N dimensional space. But you may still choose $q$s and $\dot q$s for the 6N dimensional space. The reason people use qs and ps instead, is that Hamilton's equations take more symmetrical forms in terms of these variables.