Why is the Hamiltonian a function of p and q, not p and q-dot?

[kq6up]

If the hamiltonian is defined as $\mathcal{H}\equiv \sum _{ i }^{ all }{ p_{ i }\dot { q } _{ i } } - \mathcal{L}$, how is it considered a function of $p$ and $q$ instead of $p$ and $\dot{q}$?

Chris

 

[Matterwave]

You should make a change of variables at the end to find $\dot{q}$ in terms of $p,q$.