Action integral ($I_j=\frac{1}{2\pi }{\oint }_{{\gamma }_j}\lambda$)

Near a regular Lagrangian torus, choose a one-form $\lambda$ with $d\lambda =-\omega$ and a basis of fibre cycles ${\gamma }_j$. The action $I_j=(2\pi {)}^{-1}{\int }_{{\gamma }_j}\lambda$ has differential of a smooth map $d{I}_j=(2\pi {)}^{-1}{\sum }_i{t}_i^{(j)}d{F}_i$, where $t^{(j)}$ is the corresponding joint-flow period. Nonsingularity of the period matrix makes these actions local coordinates transverse to the tori.