About the geometry of completely integrable systems

2011-05-03T18:34:01Z

During a conversation I heard an assertion that I found at least dubious for the lack of adeguate hypothesis, but I am not able to imagine a counterexample, even if it is probably obvious to some of you.
My question: is there someone who can point out to me a counterexample for the following implication?

This is the setting:
Let $(M,\omega)$ be a $2n$-dimensional symplectic manifold, and $f_1,\ldots,f_n$ independent functions on $M$ mutually Poisson-commuting, and such that the hamiltonian vector fields $X_{f_1},\ldots,X_{f_n}$ are complete.
Let $\mathcal{F}$ denote the lagrangian foliation of $M$ determined by the integrable distribution $D$ generated by $X_{f_1},\ldots,X_{f_n}$.

This is the conclusion that I find not well justified:
For any $x$ in $M$ there exists a local manifold $\Sigma_x$ which is lagrangian, transversal to $D$, and doesn't intersect any leaf of $M$ at two distinct points.
(I know that this condition is necessary and sufficient for the existence of the manifold of the leaves of $\mathcal{F}$)

Answer by DamienC for About the geometry of completely integrable systems

2011-05-03T19:10:57Z

preliminary remark

I assume that being independant for functions here means that their differentials at any point are linearly indenpendant (and not at almost any point, like it is sometimes assumed... in which case a counter-example is easy to find for $n=2$: take $f_1$ to be the square of the norm on $\mathbb{R}^2$, and look at the origin).

the statement

I claim that one can prove the following weak version of the action-angle coordinate Theorem, under the hypothesis of the question:

For any $x\in M$ there exists Darboux coordinates $(p_1,\dots,p_n,q_1,\dots, q_n)$
around $x$ such that the leaves of the Lagrangian foliation are $(q_1,\dots,q_n)=cst$.

Then the local manifold $\{p_1=\dots=p_n=0\}$ satisfies your requirement.

To prove the statement, consider $q_i=f_i-f_i(x)$ and extend them to a Darboux chart $(q_1,\dots,q_n,p_1,\dots,p_n)$ around $x$.

source of confusion

foliations may be very wilde... but here we actually have a submersion.
in usual action-angle coordinates Theorem on needs some properness assumption. But the usual Theorem tell us about properties of semi-global coordinates. Here we were dealing with a purely local statement and we don't need any properness assumption.

About the geometry of completely integrable systems - MathOverflow

About the geometry of completely integrable systems

Answer by DamienC for About the geometry of completely integrable systems

preliminary remark

the statement

source of confusion