If I have a generating function, say, $$G(q,p,P,Q)= qp - e^Q e^P\tag{1}$$

what are the equations that give me the transformations $Q=Q(p,q)$ and $P=P(q,p)$?

I have only seen generating functions depending on only two variables: $(q,Q)$, $(q,P)$, $(p,Q)$ and $(p,P)$, these cases I know are related by a Legendre transformation and then it is straightforward to get the canonical transformations. But what changes when we allow the generating function to depend on more than two variables (disconsidering time $t$)?


FWIW, in the time-independent case, besides OP's equation (1), the equations are $$ p\mathrm{d}q-P\mathrm{d}Q~=~\mathrm{d}G $$ together with $$ Q~=~Q(q,p)\qquad\text{and}\qquad P~=~P(q,p).$$

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