Goldstein et al's Classical Mechanics states that:
The Hamiltonian $H(q,p,t)$ is generated by the Legendre transformation $$ H(q,p,t) = \dot{q}_i p_i - L(q,\dot{q},t). \tag{8.15} $$
But I don't see how this equation even makes sense: $\dot q_i$, or $\dot q$ doesn't appear in the arguments of $H(q,p,t)$. In order to know $\dot q$, one would have to know not the value of $q$ but the entire function $q(t)$, and as far as I understand, $H$ is a function of the values of $q$ at a particular point in time, not the entire path over time.
This makes me doubt whether I understand what this equation means, or what the Legendre transform is.
