I'm watching a lecture on the Hamiltonian and can't figure out something. Here it is. Take a generic function G, and differentiate it with respect to p and q. What you get is the partial of G with respect to p TIMES the derivative of p (or p-dot), plus the derivative of G with respect to q TIMES q-dot. My question is, where does the p-dot and q-dot terms come into the equation here? Why isn't it just the partial of G over p plus the partial of G over q?
What you've described looks like taking the derivative of G(p,q) with respect to t, using the chain rule: $$\frac{dG(p,q)}{dt} = \frac{\partial G}{\partial p} \frac{dp}{dt} + \frac{\partial G}{\partial q} \frac{dq}{dt}$$