\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
We parameterize the Lagrangian \(L\) by with the map
\begin{align*}
\li: Q\into& T^*Q &&
q\mapsto&(q, \alpha(q))
\end{align*}
We now wish to show that \(i^*\omega=0\).
Let \((q_1, \ldots q_n)\) be local coordinates on \(Q\), so that \(\alpha(q)=\sum_{k=1}^n \alpha_k(q)dq_{k}\).
Let \(\partial_{q_i}, \partial_{q_j}\) be two basis vectors for the tangent space of \(Q\).
Let \(\{(\partial_{q_i}, 0)\}\cup\{(0, \partial p_i)\}\) be a basis for the tangent space of \(T^*Q\).
\begin{align*}
\li^*\omega(\partial_{q_i}, \partial_{q_j})=& \omega \left(\left(\partial_{q_i}, \sum_{k=1}^n (\partial_{q_i}\alpha_k )\cdot \partial_{p_k}\right), \left(\partial_{q_j}, \sum_{k=1}^n (\partial_{q_j} \alpha_k)\cdot \partial_{p_k}\right)\right)\\
=&\left( \sum_{l=1}^n dp_l \wedge dq_l\right) \left(\left(\partial_i, \sum_{k=1}^n (\partial_{q_i}\alpha_k )\cdot \partial_{p_k}\right), \left(\partial_{q_j}, \sum_{k=1}^n (\partial_{q_j} \alpha_k )\cdot\partial_{p_k}\right)\right)\\
\end{align*}
Since \(dq_i\partial_{q_j}=\delta_{ij}\) and \(dp_i\partial_{p_j}=\delta_{ij}\)
\begin{align*}
=& \partial_{q_j} \alpha_i - \partial_{q_i} \alpha_j\\
=& d\alpha(\partial_{q_i}, \partial_{q_j})
\end{align*}
This vanishes for all \(i, j\) if and only if \(\alpha\) is closed.