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For the graph embedding $\iota (q)=(q,p(q))$, pull back the symplectic form in Darboux coordinates:The coordinate two-forms are linearly independent. The graph has dimension $n$, so it is a Lagrangian submanifold exactly when this pullback vanishes, equivalently when its displayed coefficient matrix is symmetric.

Let ${\theta }_L={\sum }_i{p}_i(q)d{q}^i$. Then $d{\theta }_L=-{\iota }^{\ast }\omega$, so the same condition says that ${\theta }_L$ is a closed differential form. On a sufficiently small contractible coordinate neighborhood the Poincaré lemma supplies $S$ with ${\theta }_L=dS$. ConsequentlyOne explicit primitive on a star-shaped neighborhood of $q_0$ is $S(q)=S(q_0)+{\int }_0^1{p}_i(q_0+t(q-q_0))(q^i-q_0^i)dt$. Differentiate it and use ${\partial }_j{p}_i={\partial }_i{p}_j$ to recognize the derivative of $t{p}_j(q_0+t(q-q_0))$; the result is ${\partial }_{j}S=p_j(q)$. Conversely $p=dS$ has symmetric mixed derivatives and gives a Lagrangian graph. This construction requires the local projection to the $q$ coordinates to be nonsingular. A general Lagrangian submanifold need not be such a graph in a preassigned chart, and a closed one-form on a larger domain need not be globally exact.