Liouville–Arnold action–angle theorem

Write $\pi =F{\mathrm{\mid }}_U$ and shrink $B$ around $b_0=F(N)$ so that its closure lies in the original ball. Properness makes every fibre compact (indeed ${\pi }^{-1}(\bar{B})$ is compact). For $\alpha \in T_b^{\ast }B$ and $m\in {\pi }^{-1}(b)$, nondegeneracy defines a unique vector $X_{\alpha }(m)$ by ${\iota }_{X_{\alpha }}\omega ={\pi }^{\ast }\alpha$ at $m$: it is vertical because the fibre is Lagrangian, and the resulting map $T_b^{\ast }B\to T_m{\pi }^{-1}(b)$ is an isomorphism by dimension. In the coordinate coframe $d{F}_i$, these are constant linear combinations of the commuting $X_{F_i}$. By [F3] they are complete on each compact fibre, so their commuting flows give a smooth fibrewise $T_b^{\ast }B$-action. Its infinitesimal generators span each fibre, hence [F1] makes the action transitive.

Projection coordinates from [F3] give a local section $\sigma :B\to U$ through a chosen point of $N$. The action map $a:T^{\ast }B⟶U,a({\alpha }_b)={\alpha }_b\cdot \sigma (b),$ has invertible differential at every point: its base component is the identity and its vertical derivative is the infinitesimal-action isomorphism from step 1.1. Hence [F5] makes $a$ a local diffeomorphism. The stabilizer union $\mathrm{\Lambda }=a^{-1}(\sigma (B))$ is consequently locally a smooth section of $T^{\ast }B\to B$ near each of its points. Choose a $\mathbb{Z}$-basis of the full lattice ${\mathrm{\Lambda }}_{b_0}$ supplied by [F1]; the corresponding local sheets extend it to smooth one-forms ${\beta }_1,\dots ,{\beta }_n$ after shrinking $B$.

These continued periods generate the full lattice on every sufficiently nearby fibre. Indeed, trivialize $T^{\ast }B$ and suppose local generation fails. For each positive integer $r$, the ball of radius $1\mathrm{/}r$ about $b_0$ then contains a point $b_r$ and a period outside ${\sum }_i\mathbb{Z}{\beta }_i(b_r)$; use [A1] to choose one such pair for every $r$. Subtract integer combinations of the ${\beta }_i(b_r)$ to obtain a nonzero period ${\gamma }_r$ in their closed fundamental parallelepiped. The union of these parallelepipeds over a compact smaller ball is compact, so a convergent subsequence has limit ${\gamma }_0\in {\mathrm{\Lambda }}_{b_0}$ by continuity of $a$ and $a({\gamma }_r)=a(0_{b_r})=\sigma (b_r)$. Write ${\gamma }_0={\sum }_i{m}_i{\beta }_i(b_0)$ and replace ${\gamma }_r$ by ${\delta }_r={\gamma }_r-{\sum }_i{m}_i{\beta }_i(b_r)$. Then every ${\delta }_r$ is a nonzero stabilizer and ${\delta }_r\to 0_{b_0}$. But [F5] makes $a$ injective on one neighbourhood of $0_{b_0}$, while $a({\delta }_r)=a(0_{b_r})$ and both arguments eventually lie there, a contradiction. Equivalently, on a compact smaller base one may cover the zero section by finitely many such inverse-function neighbourhoods to obtain a uniform zero-free fibre neighbourhood. Thus ${\beta }_1(b),\dots ,{\beta }_n(b)$ are a full smooth period-lattice basis.

If $\alpha \in {\mathrm{\Omega }}^1(B)$, the flow ${\mathrm{\Phi }}_{\alpha }^t$ of $X_{\alpha }$ is vertical. By [F4], $\frac{d}{dt}({\mathrm{\Phi }}_{\alpha }^t{)}^{\ast }\omega =({\mathrm{\Phi }}_{\alpha }^t{)}^{\ast }{\pi }^{\ast }d\alpha ={\pi }^{\ast }d\alpha$, where the last equality uses $\pi \circ {\mathrm{\Phi }}_{\alpha }^t=\pi$. Hence $({\mathrm{\Phi }}_{\alpha }^1{)}^{\ast }\omega =\omega +{\pi }^{\ast }d\alpha$. For a sheet ${\beta }_i$ of $\mathrm{\Lambda }$, ${\mathrm{\Phi }}_{{\beta }_i}^1$ is the identity on every fibre, so injectivity of pullback by the submersion gives $d{\beta }_i=0$.

By [F4], ${\beta }_i=d{I}_i$ after shrinking the ball. The ${\beta }_i(b)$ form a vector-space basis by step 3.1, so [F5] makes $(I_1,\dots ,I_n)$ a coordinate system after one further shrink. The fibre action modulo the now-proved full lattice is a free transitive $(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^n$-action; write its period-one coordinates as ${\theta }_i$.

Start with any local section $\sigma$. Its pullback $\tau ={\sigma }^{\ast }\omega$ is closed. On the ball [F4] gives $\tau =d\alpha$. The translated section ${\sigma }^{\mathrm{\prime }}={\mathrm{\Phi }}_{-\alpha }^1\circ \sigma$ satisfies $({\sigma }^{\mathrm{\prime }}{)}^{\ast }\omega =\tau -d\alpha =0$ by step 4.1, so it is Lagrangian.

Acting on ${\sigma }^{\mathrm{\prime }}$ gives a diffeomorphism $B\times (\mathbb{R}\mathrm{/}\mathbb{Z}{)}^n\to U$: it is fibrewise bijective by transitivity and the stabilizer lattice, and locally a diffeomorphism by step 2.1. Its vertical coordinate vector ${\mathrm{\partial }}_{{\theta }_i}$ maps to $X_{d{I}_i}$. Thus, for every base tangent $v$, $\omega (X_{d{I}_i},v)=d{I}_i(v)$. Both $\omega$ and ${\sum }_{i}d{\theta }_i\wedge d{I}_i$ vanish on vertical pairs. Their horizontal--horizontal evaluations vanish on the zero-angle section by step 5.2 and hence everywhere, because fixed-angle translations are symplectic by step 4.1. These evaluations exhaust all tangent pairs, proving $\omega ={\sum }_{i}d{\theta }_i\wedge d{I}_i$.

Since $F$ is constant on each fibre and $I$ are coordinates on the base, each $F_i$ and every other fibre-constant Hamiltonian is a function of $I$. Beyond the inherited compact-flow uses and the countable counterexample sequence in step 3.1, only one section, a finite lattice basis, and primitives on one ball were selected.