Let $M$ be a smooth finite-dimensional second-countable Hausdorff manifold, let $\omega\in\Omega^2(M)$ be a closed nondegenerate $2$-form, and let $L\subset M$ be an embedded submanifold such that, for every $x\in L$, the subspace $T_xL\subset T_xM$ is Lagrangian with respect to $\omega_x$. Let $\lambda\in\Omega^1(T^*L)$ denote the tautological one-form on $T^*L$, and let

\begin{align*} \omega_{\mathrm{can}}=-d\lambda \end{align*}

be the canonical symplectic form on $T^*L$. Then there exist an open neighbourhood $U\subset T^*L$ of the zero section, an open neighbourhood $V\subset M$ of $L$, and a diffeomorphism

\begin{align*} \Phi:U\to V \end{align*}

such that, after identifying $L$ with the zero section of $T^*L$,

\begin{align*} \Phi|_L=\operatorname{id}_L \end{align*}

and

\begin{align*} \Phi^*\omega=\omega_{\mathrm{can}}|_U. \end{align*}