Hamilton's Equations and Generating Function

[Tangent87]

Say we have a Hamiltonian H(q,p,t) and we then transform from p and q to P=P(q,p,t) and Q=Q(q,p,t), with:

P\dot{Q}-K=p\dot{q}-H+\frac{d}{dt}F(q,p,Q,P,t)

where K is the new Hamiltonian. How do we show that P and Q obey Hamilton's equations with Hamiltonian K? I have tried partial differentiating both sides of the above w.r.t Q and P but I'm not sure what to differentiate (i.e. do we consider p and q independent from P and Q?). I also expanded the big dF/dt but it didn't seem to help.

 

[Tangent87]

Anybody? I can rewrite the equation as

P\dot{Q}-K=p\dot{q}-H+\dot{q}\frac{\partial F}{\partial q}+\dot{p}\frac{\partial F}{\partial p}+\dot{q'}\frac{\partial F}{\partial q'}+\dot{p'}\frac{\partial F}{\partial p'}+\frac{\partial F}{\partial t}

It doesn't seem to help me as I don't know whether to just partial differentiate both sides w.r.t p' and q' in an attempt to find \frac{\partial K}{\partial p'} and \frac{\partial K}{\partial q'} or if there's something more subtle needed?