= Thermodynamic potential as a Lagrangian graph
{title2=$L_M=\operatorname{graph}(dM)$}
For a smooth function $M(q)$, its differential defines a <Lagrangian submanifold> of the <cotangent bundle>: pullback of the <canonical one-form on a cotangent bundle> is $dM$, so pullback of the <symplectic form> $-d\theta$ is zero. The graph has half the ambient dimension. With $q=(S,V,N)$ and momentum coordinates $(T,-P,\mu)$, the <first law of thermodynamics> identifies the equilibrium graph with $dM$. This geometric statement does not require extensivity or thermodynamic stability.
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