Canonical transformations, generating function

[fluidistic]

Homework Statement

Given the generating function F=\sum _i f_i (q_j,t)P_i,
1)Find the corresponding canonical transformations.
2)Show that the transformations of generalized coordinates are canonical transformations.
3)What meaning does the canonical transformation originated by the generating function \Phi (q,P)=\alpha qP has?

Homework Equations

p_i=\frac{\partial F }{\partial q_i}, P_i=-\frac{\partial F }{\partial Q_i}, H'=H+\frac{\partial F }{\partial t}.

The Attempt at a Solution

I don't know how to start. The notation confuses me, particularly the j. Should the sum be a sum over i and j?

Edit:1)Q_i=\frac{\partial F }{\partial P_i}=f_i(q_j,t). A canonical transformation is such that \dot Q_i=\frac{\partial H'}{\partial P_i} and \dot P_i =-\frac{\partial H'}{\partial Q_i}.
Therefore I guess I must verify that \frac{\partial ^2 F}{\partial t \partial P_i}=\dot Q_i and that... oh well I'm totally confused.

 

[member 553083]

2)This is not really clear to me. I mean, I know that canonical transformations preserve the form of the Hamiltonian equations of motion, but how does this relate to the generating function?3)I'm guessing this means that the momentum and position are related by an exponential function, but I'm not sure.