This is probably a very elementary question in symplectic geometry, a subject I've picked up by osmosis rather than ever really learning.


Suppose I have a symplectic manifold $M$. I believe that a *Lagrangian fibration* of $M$ is a collection of immersed Lagrangian submanifolds so that as a fibered manifold locally $M$ looks like a product. I.e. I can find local coordinates so that the fibers are $\{\text{half the coordinates} = \text{constant}\}$.


Then, at least locally, I can think about the set of fibers as some sort of "space" $N$. My question is: to what extent can I think of $M$ as the cotangent bundle $T^\*N$?


Surely the answer is "to no extent whatsoever" globally: the set of fibers is probably not a space in any good way, and certainly not a manifold (see: irrational line in a torus). But what about locally? Then it's really two questions:



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> **Question 1:** If I have a lagrangian fibration in $M$, can I find local coordinates $p\_i,q^j: M \to \mathbb R$ so that the symplectic form is $\omega = \sum\_i dp\_i \wedge dq^i$ and the fibers are of the form $\{ \vec q = \text{constant}\}$.
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I thought the answer was obviously "yes", and maybe it is, but what I thought worked I can't make go through all the way.


Then the question is about how canonical this is, and that's not really about general Lagrangian fibrations at all:



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> **Question 2:** What is a good description of the local symplectomorphisms $\mathbb R^{2n} \to \mathbb R^{2n}$ of the form $\tilde q = \tilde q(q)$ and $\tilde p = \tilde p(p,q)$?
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The beginning of the answer is that it is a local symplectomorphism if $\sum\_i d\tilde p\_i \wedge d\tilde q^i = \sum\_i dp\_i \wedge dq^j$, but the left-hand-side is $\sum\_{i,j,k} \bigl(\frac{\partial \tilde p\_i}{\partial q^j}dq^j + \frac{\partial \tilde p\_i}{\partial p\_j}dp\_j \bigr) \wedge \bigl( \frac{\partial \tilde q^i}{\partial q^k}dq^k \bigr)$, so the two conditions are that $\sum\_i \frac{\partial \tilde p\_i}{\partial q^j} \frac{\partial \tilde q^i}{\partial q^k}dq^k$ is a symmetric matrix, and that $\frac{\partial \tilde p}{\partial p}$ is the (maybe transpose, depending on your convention) inverse matrix to $\frac{\partial \tilde q}{\partial q}$.


Anyway, I guess for completeness I'll also ask the global question:



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> **Question 0:** What global conditions on $M$ and the fibration assure that there is a global symplectomorphism with $T^\*N$ for some $N$?
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To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?
Your Question 1 is called Darboux theorem for fibrations (see: Arnold, V., Givental, A., Symplectic geometry, Dynamical Systems IV, Symplectic Geometry and its Applications (Arnold, V., Novikov, S., eds.), Encyclopaedia of Math. Sciences 4, Springer-Verlag, Berlin-New York, 1990.)


Here is how to construct suitable Darboux coordinates. Let $q\_i$ be local coordinates in the base of the fibration, we identify them with their pullbacks to the symplectic manifold. The functions $q\_i$ generate Hamiltonian vector fields $X\_{i}$ and these fields are tangent to the fibers (note that $X\_{i}$'s commute). Let $\varphi\_{i}(t)$ be the flow map generated by $X\_{i}$ for time $[0,t]$. 


Now we choose (locally) a Lagrangian submanifold $L$ transversal to the fibration. The coordinates $q\_i$ give coordinates on $L$, so $(q\_1,...,q\_n)$ stands for a point on $L$. Here is a construction of a local symplectomorphism $$(p\_1,...,p\_n,q\_1,...,q\_n) \mapsto \varphi\_{n}(p\_n)\circ ...\circ \varphi\_{1}(p\_1)(q\_1,...,q\_n).$$ It is easy to check that it is indeed a fibered symplectomorfism sending the symplectic structure to the standard one.


