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Assume that we have one of the 4 types of generating functions (say F(p, Q)) which defines a symplectomorphism $T$: $(P,Q) = (u(p,q), v(p,q))$. Now we can make a substitution and get a function $G(p,q)$ of old variables $(p,q)$ by formula $G(p,q)= F(p, v(p,q))$.

Questions:

  1. Is there a specific geometric meaning for $G$?
  2. Does $G$ uniquely define $T$ if we know the type (1,2,3,4) of $F$? If not, what is the ambiguity?
  3. Are $G$-s of 4 types related to each other?
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