# Canonical transformations, generating function

1. Feb 29, 2012

### fluidistic

1. The problem statement, all variables and given/known data
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?

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

3. 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.

Last edited: Feb 29, 2012