= Lagrangian graph
{c}
{title2=$\operatorname{graph}(\alpha)\subset T^*Q$}
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 $p_i=\partial_iS$. Globally a closed one-form may have nonzero periods and no single generating function.
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