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To make my explanation easier open the ''Generating function approach'' section on this wiki article:
http://en.wikipedia.org/wiki/Canonical_transformation
The function ##\frac{dG}{dt}## represents the function that always can be added to the Lagrangian without changing the mechanical equations.
However, in the Lagrangian mechanics formulation the function ##G## is only allowed to be a function of the coordinates and time ##G(q,t)##.
If I look further down in the wiki article I find that ##G## is allowed to have other variables like ##G(p,Q,t)##. It isn't very clear why it is trivial or easy to see that this won't give any problems.
http://en.wikipedia.org/wiki/Canonical_transformation
The function ##\frac{dG}{dt}## represents the function that always can be added to the Lagrangian without changing the mechanical equations.
However, in the Lagrangian mechanics formulation the function ##G## is only allowed to be a function of the coordinates and time ##G(q,t)##.
If I look further down in the wiki article I find that ##G## is allowed to have other variables like ##G(p,Q,t)##. It isn't very clear why it is trivial or easy to see that this won't give any problems.