Lagrangian graph ($\mathrm{graph}(\alpha )\subset T^{\ast }Q$)

The graph of a one-form $\alpha$ on a smooth manifold $Q$ is a Lagrangian submanifold of its cotangent bundle exactly when $d\alpha =0$. Indeed the graph pulls the tautological one-form back to $\alpha$, and therefore pulls the canonical symplectic form back to $-d\alpha$. Locally the Poincaré lemma writes $\alpha =dS$, giving $p_i={\partial }_{i}S$. Globally a closed one-form may have nonzero periods and no single generating function.