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It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on.

Now, I stumbled over the question whether in some particular cases, as I will describe below, local symplectomorphisms are indeed global ones.

The question arouse in the context of action-angle variables. I was reading these lecture notes click me and go to page 12 of the PDF or 11 of the notes.

Then the new action angles are constructed as derivatives of the generating function with respect to $I_1,...,I_n.$ But of course, since action-angles are periodic, this derivative is not well-defined globally, still one can consider this action locally and everything works fine. In this case, one can locally say that since the generating function is locally well-defined, we induce a symplectomorphism $(q,p) \mapsto (I,\phi)$.

The author does not really adress this local/global issue, probably because it does not matter apparently. So why does a local generating function induce a global symplectomorphism?

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Local symplectomorphism, in general, do not induce global ones. In fact every symplectic manifold is locally symplectomorphic to to the standard $(\mathbb R^{2n},\omega_{n})$ (Darboux theorem) while not (even if diffeomorphic to it) not globally symplectomorphic to it.

Even in the context of action-angle variables one does not have a global symplectomorphism. The issue is relevant and matters, Probably it is not treated in those notes just because it requires different techniques.

The theorem you refer to is often cited in the literature as "local Arnold-Liouville" theorem, to emphasize the local nature both of the action-angle variables, which, as you say, are not globally defined, and of the symplectomorphism. The theorem, as stated, holds in a neighbourhood of a Lagrangian torus in $M$. More generally an integrable system may be seen as a Lagrangian fibration $\pi:M\to B$ with compact connected Lagrangian fibers. The existence of a global symplectomorphism as you seem to wish is basically equivalent to a splitting of such fibration, i.e. to the existence of a global Lagrangian section. No surprise that such existence is possible only if some topological obstructions (Chern class) are zero.

The theory of global action-angle coordinates was developed by Duistermaat "On global action-angle variables" Commun. Pure Appl. Math. 33, 687-706 (1980). I personally found these notes quite helpful in getting a grasp on it

http://people.sissa.it/~bruzzo/gaav/gaav.pdf

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  • $\begingroup$ thank you. I also asked another question on this topic covered in Arnold's book. mathoverflow.net/questions/218137/… . Maybe you want to take a look at it. $\endgroup$
    – QuantumTheory
    Commented Sep 13, 2015 at 11:47
  • $\begingroup$ it is also interesting in this context, to hear your opinion on this, because in your reference, the author says that we have this coordinate system on suitabe open subsets of $M$, whereas Arnold claims the existence in the nbh of the reduced manifold (the torus) $\endgroup$
    – QuantumTheory
    Commented Sep 13, 2015 at 11:59

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