## Formalized Name Action-Angle Coordinates Near a Compact Regular Fiber ## Formalized Statement Let $(M,\omega)$ be a smooth symplectic manifold of dimension $2n$, and let $F=(f_1,\dots,f_n):M\to\mathbb R^n$ be a smooth map such that the functions $f_i$ Poisson commute and $dF_x:T_xM\to\mathbb R^n$ has rank $n$ at every point under consideration. Let $c\in\mathbb R^n$ be a regular value of $F$, and suppose that the fiber $L=F^{-1}(c)$ is compact and connected. Then there are an open neighbourhood $W\subset\mathbb R^n$ of $c$, the connected component $V\subset F^{-1}(W)$ containing $L$, an open set $U\subset\mathbb R^n$, and a diffeomorphism $\Phi:V\to U\times\mathbb T^n$ with coordinate functions $(I_1,\dots,I_n,\theta_1,\dots,\theta_n)$ such that $F|_V:V\to W$ is a smooth fibration by compact connected Lagrangian tori, \begin{align*} \omega=\sum_{i=1}^n dI_i\wedge d\theta_i. \end{align*} Moreover, for each $j\in\{1,\dots,n\}$ there exists a smooth function $h_j:U\to\mathbb R$ such that \begin{align*} f_j=h_j(I_1,\dots,I_n) \end{align*} on $V$. ## Proof [proofplan] We first shrink around the compact regular fiber so that the restriction of $F$ is a proper submersion, hence a smooth fibration. The commuting Hamiltonian vector fields span each nearby fiber, so every fiber is a homogeneous quotient of $\mathbb R^n$ by a full period lattice and is therefore a torus; the same Poisson-commuting condition makes the fibers Lagrangian. Finally we apply the standard Darboux theorem for Lagrangian torus fibrations with a smooth section, which gives action-angle coordinates compatible with the given fibration, and this compatibility implies that the original integrals depend only on the action variables. [/proofplan] [step:Shrink to a proper regular fibration near the compact fiber] Choose an open neighbourhood $W_0\subset\mathbb R^n$ of $c$ consisting of regular values of $F$ on the component of $F^{-1}(W_0)$ containing $L$. This is possible because $dF_x$ has rank $n$ along $L$ and rank is an open condition. Let $V_0\subset F^{-1}(W_0)$ denote that component. Then \begin{align*} F|_{V_0}:V_0\to W_0 \end{align*} is a smooth submersion. We use the standard local properness lemma for submersions near a compact fiber: if $p:N\to B$ is a smooth submersion and $p^{-1}(b_0)$ is compact, then, after shrinking to an open neighbourhood $B_1\subset B$ of $b_0$, the component of $p^{-1}(B_1)$ containing $p^{-1}(b_0)$ maps properly to $B_1$. Applying this lemma to $F|_{V_0}$ and $L=(F|_{V_0})^{-1}(c)$, choose a connected contractible open neighbourhood $W\subset W_0$ of $c$ such that the connected component \begin{align*} V\subset F^{-1}(W) \end{align*} containing $L$ maps properly to $W$. Since $F|_V$ is both a proper smooth submersion and a submersion onto $W$, Ehresmann's fibration theorem gives that \begin{align*} F|_V:V\to W \end{align*} is a smooth locally trivial fibration with compact fibers. For $b\in W$, define \begin{align*} L_b:=F^{-1}(b)\cap V. \end{align*} After replacing $W$ by the image of the fibration component if necessary, all fibers $L_b$ are connected because $W$ is connected and the number of connected components of the fibers in a proper locally trivial fibration is locally constant. [guided] The first task is not to construct coordinates, but to isolate the part of the system where no singular behavior occurs. Because $c$ is a regular value and $L=F^{-1}(c)$ is compact, every point of $L$ has a neighbourhood on which $dF$ has rank $n$. Finitely many such neighbourhoods cover $L$, and shrinking the target around $c$ gives an open set $W_0\subset\mathbb R^n$ such that the component $V_0\subset F^{-1}(W_0)$ containing $L$ consists only of regular points. Therefore \begin{align*} F|_{V_0}:V_0\to W_0 \end{align*} is a smooth submersion. A submersion alone need not be a fibration, because fibers can escape to infinity. The compactness of the reference fiber prevents this after shrinking the base. We invoke the standard local properness lemma for submersions near a compact fiber: if $p:N\to B$ is a smooth submersion and $p^{-1}(b_0)$ is compact, then there is a neighbourhood $B_1$ of $b_0$ such that the connected component of $p^{-1}(B_1)$ containing $p^{-1}(b_0)$ is proper over $B_1$. Applying this to $p=F|_{V_0}$ and $b_0=c$, we shrink to a connected contractible neighbourhood $W\subset W_0$ and let $V\subset F^{-1}(W)$ be the component containing $L$. Then \begin{align*} F|_V:V\to W \end{align*} is a proper smooth submersion. Ehresmann's fibration theorem applies exactly to proper smooth submersions. Hence $F|_V$ is a smooth locally trivial fibration. Its fibers are compact because the map is proper. For each $b\in W$ we write \begin{align*} L_b:=F^{-1}(b)\cap V. \end{align*} Since $W$ is connected and a locally trivial fibration has locally constant fiber type, the number of connected components of $L_b$ is independent of $b$. At $b=c$ the fiber is $L$, which is connected by hypothesis, so every nearby fiber $L_b$ is connected. [/guided] [/step] [step:Show that the nearby fibers are Lagrangian tori] For each $j\in\{1,\dots,n\}$, let $X_j\in\mathfrak X(V)$ be the Hamiltonian vector field defined by \begin{align*} \iota_{X_j}\omega=df_j. \end{align*} For every $i,j\in\{1,\dots,n\}$, \begin{align*} df_i(X_j)=\omega(X_i,X_j)=\{f_i,f_j\}=0. \end{align*} Thus each $X_j$ is tangent to every fiber $L_b$. Since $df_1,\dots,df_n$ are linearly independent on $V$ and $\omega$ is nondegenerate, the vector fields $X_1,\dots,X_n$ are pointwise linearly independent. The identity $[X_i,X_j]=X_{\{f_i,f_j\}}$ for Hamiltonian vector fields gives $[X_i,X_j]=0$. On a compact fiber $L_b$, the restrictions $X_1|_{L_b},\dots,X_n|_{L_b}$ are complete and commute, so they integrate to an action \begin{align*} \rho_b:\mathbb R^n\times L_b\to L_b. \end{align*} The infinitesimal action is injective at every point, so each orbit has dimension $n$. Since $\dim L_b=2n-n=n$, every orbit is open in $L_b$. The connectedness of $L_b$ implies that the action is transitive. Fix $x\in L_b$ and define the stabilizer subgroup \begin{align*} \Lambda_{b,x}:=\{t\in\mathbb R^n:\rho_b(t,x)=x\}. \end{align*} The injectivity of the infinitesimal action implies that $\Lambda_{b,x}$ is discrete. The orbit map induces a diffeomorphism \begin{align*} \mathbb R^n/\Lambda_{b,x}\to L_b. \end{align*} Since $L_b$ is compact, the quotient $\mathbb R^n/\Lambda_{b,x}$ is compact, so $\Lambda_{b,x}$ is a full lattice in $\mathbb R^n$. Hence $L_b$ is diffeomorphic to $\mathbb T^n$. Finally, for $u,v\in T_xL_b$, write \begin{align*} u=\sum_{i=1}^n a_iX_i(x) \end{align*} and \begin{align*} v=\sum_{j=1}^n b_jX_j(x) \end{align*} with $a_i,b_j\in\mathbb R$. Then \begin{align*} \omega_x(u,v)=\sum_{i=1}^n\sum_{j=1}^n a_i b_j\omega_x(X_i(x),X_j(x))=\sum_{i=1}^n\sum_{j=1}^n a_i b_j\{f_i,f_j\}(x)=0. \end{align*} Thus $L_b$ is isotropic of dimension $n$ in the $2n$-dimensional symplectic manifold $(M,\omega)$, hence $L_b$ is Lagrangian. [/step] [step:Choose a global section and smooth period basis] Since $F|_V:V\to W$ is a smooth torus bundle and $W$ is contractible, the bundle is smoothly trivial after possibly shrinking $W$. In particular, there is a smooth section \begin{align*} s:W\to V \end{align*} with $F\circ s=\operatorname{id}_W$. For $b\in W$, define the period lattice at the section point by \begin{align*} \Lambda_b:=\{t\in\mathbb R^n:\rho_b(t,s(b))=s(b)\}. \end{align*} The standard period-lattice regularity theorem for free transitive commuting flows on a smooth torus fibration says that the union \begin{align*} \Lambda:=\{(b,t)\in W\times\mathbb R^n:t\in\Lambda_b\} \end{align*} is a smooth rank-$n$ lattice subbundle of the trivial bundle $W\times\mathbb R^n\to W$. Its hypotheses are satisfied here because the fields $X_1,\dots,X_n$ are smooth, commute, are tangent to the fibers, are pointwise linearly independent on the fibers, and their induced $\mathbb R^n$-action is transitive with compact torus orbits. Since $W$ is contractible, the lattice bundle has a smooth global basis. Thus there are smooth maps \begin{align*} \tau_i:W\to\mathbb R^n \end{align*} for $i\in\{1,\dots,n\}$ such that \begin{align*} \Lambda_b=\mathbb Z\tau_1(b)+\dots+\mathbb Z\tau_n(b) \end{align*} for every $b\in W$. [/step] [step:Apply the Lagrangian fibration normal form to obtain compatible action-angle coordinates] We now invoke the standard Darboux theorem for Lagrangian fibrations with section. In the form needed here, it states: if $\pi:E\to B$ is a smooth fibration whose fibers are compact Lagrangian tori in a symplectic manifold $(E,\Omega)$, if $B$ is contractible, and if $\pi$ has a smooth section, then after shrinking $B$ there are an open set $U\subset\mathbb R^n$, a diffeomorphism \begin{align*} \Phi:E\to U\times\mathbb T^n, \end{align*} and coordinates $(I_1,\dots,I_n,\theta_1,\dots,\theta_n)$ such that the fibers of $\pi$ are exactly the sets on which $I=(I_1,\dots,I_n)$ is constant and \begin{align*} \Omega=\sum_{i=1}^n dI_i\wedge d\theta_i. \end{align*} We apply this theorem to \begin{align*} \pi=F|_V:V\to W \end{align*} and \begin{align*} \Omega=\omega|_V. \end{align*} The preceding steps verify all hypotheses: $F|_V$ is a smooth fibration, its fibers are compact Lagrangian tori, $W$ is contractible, and a smooth section $s:W\to V$ has been chosen. After shrinking $W$ and replacing $V$ by the connected component of $F^{-1}(W)$ containing $L$, the normal form gives a diffeomorphism \begin{align*} \Phi:V\to U\times\mathbb T^n \end{align*} with action-angle coordinates satisfying \begin{align*} \omega=\sum_{i=1}^n dI_i\wedge d\theta_i. \end{align*} Moreover, by the compatibility part of the normal form, each fiber $L_b$ is a common level set of $I=(I_1,\dots,I_n)$. [guided] This is the step where the ordinary Lagrangian neighbourhood theorem would not be enough. That theorem gives Darboux coordinates near one Lagrangian torus, but it does not guarantee that the coordinate tori are the original fibers of $F$. We need the stronger normal form for a whole Lagrangian torus fibration. The standard Darboux theorem for Lagrangian fibrations with section says the following. Let $\pi:E\to B$ be a smooth fibration inside a symplectic manifold $(E,\Omega)$. Assume that every fiber of $\pi$ is a compact Lagrangian torus, that $B$ is contractible, and that $\pi$ admits a smooth section. Then, after shrinking $B$, there are an open set $U\subset\mathbb R^n$, a diffeomorphism \begin{align*} \Phi:E\to U\times\mathbb T^n, \end{align*} and coordinates $(I_1,\dots,I_n,\theta_1,\dots,\theta_n)$ such that two conclusions hold at once. First, the symplectic form has the Darboux action-angle expression \begin{align*} \Omega=\sum_{i=1}^n dI_i\wedge d\theta_i. \end{align*} Second, the fibration is respected: the fibers of $\pi$ are exactly the common level sets of the action map $I=(I_1,\dots,I_n)$. We verify the hypotheses for $\pi=F|_V$ and $\Omega=\omega|_V$. The first step proved that $F|_V:V\to W$ is a smooth fibration. The second step proved that every fiber $L_b$ is a compact Lagrangian torus. We chose $W$ contractible when shrinking the base. The third step supplied a smooth section $s:W\to V$. Therefore the normal form theorem applies. Its conclusion gives an open set $U\subset\mathbb R^n$ and a diffeomorphism \begin{align*} \Phi:V\to U\times\mathbb T^n \end{align*} with coordinate functions $(I_1,\dots,I_n,\theta_1,\dots,\theta_n)$. The symplectic form is \begin{align*} \omega=\sum_{i=1}^n dI_i\wedge d\theta_i, \end{align*} and, crucially, the common level sets of $I$ are the original Liouville tori $L_b$. This compatibility is exactly what is missing from a proof that only applies the Lagrangian neighbourhood theorem to the single reference fiber. [/guided] [/step] [step:Factor the integrals through the action variables] Let \begin{align*} I:V\to U \end{align*} denote the smooth action map $x\mapsto (I_1(x),\dots,I_n(x))$. By the compatibility conclusion of the Lagrangian fibration normal form, the fibers of $I$ are the same subsets of $V$ as the fibers of $F|_V$. Hence $F$ is constant on each fiber of $I$. Define \begin{align*} h:U\to\mathbb R^n \end{align*} by declaring $h(a)=F(x)$ for any $x\in V$ with $I(x)=a$. This is well-defined because $F$ is constant on $I^{-1}(a)$. Since $I$ is the projection to the first factor in the coordinates $\Phi:V\to U\times\mathbb T^n$, smoothness of $h$ follows by evaluating $F$ on any smooth local section of that projection, for instance on a local constant-angle section. Thus \begin{align*} F=h\circ I. \end{align*} Writing $h=(h_1,\dots,h_n)$ gives, for every $j\in\{1,\dots,n\}$, \begin{align*} f_j=h_j(I_1,\dots,I_n). \end{align*} Together with the symplectic formula for $\omega$, this proves the theorem. [/step]