\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

At each \(q\in L\cap Q\), consider the Lagrangian submanifold \(T^*qQ\). Take local coordinates \((q_1, \ldots, q_n, p_1, \ldots, p_n)\) identifying \(q\) with the origin so that \(T^*qQ\) is the linear subspace in the \(p_i\) directions. We can take a Weinstein neighborhood \(T^*_\epsilon(T^*_qQ)\) of \(T^*_qQ\), whose cotangent bundle structure is \begin{align*} T^*_\epsilon(T^*_qQ)\to T^*qQ && (q_1, \ldots, q_n, p_1, \ldots, p_n)\mapsto (p_1, \ldots, p_n). \end{align*} Since the intersection \(L\cap Q\) is transverse, the projection \(T_qL\to T_q(T^*qQ)\) is surjective. Therefore when restricted to a small enough neighborhood of \(\in U\subset T^*_qQ\) the Lagrangian \(L|_{T^*_\epsilon U}\) presents itself as a section of \(T^*_\epsilon U\to U\). Therefore, there exists a one form \(\eta\in \Omega^1(U)\) so that \(L|_{T^*U}\) is parameterized by \((p, \eta_p)\). By taking an even smaller \(U\), we may assume that \(U\) is a contractible neighborhood, and \(\eta=dH\) is an exact one-form. Pick \(\rho\) a function which vanishes in a neighborhood of \(q\in U\), and takes the value \(1\) in a neighborhood of \(\partial U\). Consider the Lagrangian section of \(T^*U\) parameterized by \(d(\rho \cdot H)\). This section is Hamiltonian isotopic to \(L|_{T^*_\epsilon U}\) relative boundary. Additionally, \(d(\rho\cdot H)\) agrees with \(U=T^*_qQ\) in a small neighborhood of \(q\). The Lagrangian submanifold \(L\setminus (L|_{T^*U})\cup (d(\rho\cdot H))\) is Hamiltonian isotopic to \(L\).