If we know a generating function $\phi$ for a canonical transformation and also we know a generating function W for a Lagrangian submanifold $\Lambda$, what is a generating function for $\phi(\Lambda)$?
2026-03-25 22:09:19.1774476559
Canonical transformation, Lagrangian submanifold and generating function
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You can find an answer in the paper by Brunella $\textit{On a Theorem of Sikorav}$, Enseign. Math. (2) 37, No. 1--2, 83--87 (1991; Zbl 0739.58015). Lemma 4 is the result you are looking for, and you can find the paper here. Note that however there is the hypothesis that the symplectomorphism is compactly supported, and I do not know a reference where it is removed (nor if in fact the result is still true).