Darboux chart compatible with Lagrangian submanifold

Question

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ and let $L\subset M$ be a Lagrangian submanifold, that is $\dim L=n$ and $$\omega_p(u,v) = 0$$ for all $u,v\in T_pL$ holds for every point $p\in L.$ We wish to show that for each $p\in L,$ there exist local coordinates $x_1,\dots,x_n,y_1,\dots,y_n$ for a neighborhood $U\subset M$ of $p$ so that $$\omega = \sum_{i=1}^ndy_i\wedge dx_i$$ and $$L\cap U = \{x_1=\cdots=x_n=0\}.$$ The first condition is that the coordinates are a Darboux chart, which we know exists from Darboux's theorem. I am familiar with the proof of this theorem using linear algebra and Moser's trick, but am unsure how to adapt it so that the second condition holds. A different approach described here uses the Weinstein Lagrangian neighborhood theorem to take $M=T^*L$ without loss, in which case the usual coordinates for the cotangent bundle meet our conditions. However, I am unfamiliar with a lot of what is being used in the proofs I have found of the Lagrangian neighborhood theorem and want to find a more direct approach. Does anyone have any suggestions? Thanks!

Answer 1

Pick any local Darboux coordinates. We can suppose (after a bit of renaming) that $L$ is given in these coordinates by equations $x_i=g_i(p_1,\dots,p_n)$ for some functions $g_i$. $L$ being Lagrangian is equivalent to $\sum_i dp_i \wedge dg_i=0$. So just set $x'_i=x_i-g_i$ and you have your Darboux coordinates $x'_i,p_i$.

(As I wrote in the comment above, Weinstein LN theorem is more difficult and requires a version of the Moser's trick.)