Arnold's theorem on action-angles.

Solution 1:

Okay, I learned this from Arnold's book, but it took me years to understand it the way I do now, and I am going to explain it here this way. I shall address the Original Porster's question 4, about the global argument; however, I am not addressing the other specific questions.

The general picture is the following: Liouville theorem is a semilocal symplectic linearisation result for a special $\mathbb{R}^n$-action on a $2n$-dimensional symplectic manifold. And to understand that it helps to see Picard-Lindelöf theorem about existence and uniqueness of solutions of ordinary differential equations, and Frobenius theorem, as smooth linearisation of vector fields near points where they do not vanish (nonsingular points).

Definition: An integrable system on a $2n$-dimensional symplectic manifold $(M,\omega)$ is a mapping $F=(f_1,\dots,f_n):M\to \mathbb{R}^n$ such that:

it is a submersion on an open dense subset of $M$;

its components Poisson commute amongst each other, $\{f_j,f_k\}_\omega=0$;

the hamiltonian vector fields generated by its components are complete (some authors do not assume this condition, yet it holds in some cases, e.g. when the symplectic manifold is compact).

The classical Liouville theorem on the integrability of hamiltonian systems provides a semilocal normal form for the hamiltonian flow and symplectic form near a regular level set of its first integrals.

Theorem: Let $F=(f_1,\dots,f_n):M\to \mathbb{R}^n$ be an integrable system on a symplectic manifold $(M,\omega)$.

The hamiltonian vector fields generated by its components define an integrable (in the Sussmann sense) distribution of the tangent bundle whose leaves are generically lagrangian, with isotropic singular leaves.

The connected components of the preimage of regular values (regular leaves) are homogeneous $\mathbb{R}^n$ spaces; they are diffeomorphic to $\mathbb{R}^{n-m}\times\mathbb{T}^m$.

The foliation is a lagrangian fibration in a neighbourhood of each regular leaf; it defines a fibre bundle with lagrangian fibres.

There are coordinates on a local trivialisation of each lagrangian leaf in which $\omega$ is in Darboux form and the flows induced by each $f_j$ are linear.

In other words, the Liouville theorem gives a description of integrable systems near the regular points of the mapping $F$.

A remark is important here: I am not assuming that the first integrals are everywhere functionally linearly independent, and this is why I need to invoke Sussmann theorem. For the sake of simplicity one can just restrict to the open and dense subset where this functions are functionally linearly independent and use Frobenius theorem instead.

It is not hard to see that the following definition is equivalent to the previous one (assuming the completeness condition).

Definition': An integrable system à la Liouville on a symplectic manifold $(M,\omega)$ is a hamiltonian $\mathbb{R}^n$-action $\rho:\mathbb{R}^n\to\mathrm{Diff}(M)$, whose stabiliser subgroups are discrete over an open dense subset of $M$, together with an equivariant comomentum mapping $\mu^*:\mathbb{R}^n\to C^\infty(M;\mathbb{R})$.

Let me describe the comomentum mapping and, then, the action. Fixing a basis of $\mathbb{R}^n$, $v_1,\dots,v_n\in\mathbb{R}^n$ one can define a linear mapping $\mu^*:\mathbb{R}^n\to C^\infty(M;\mathbb{R})$ by $\mu^*(v_j):=f_j$, where $f_j\in C^\infty(M;\mathbb{R})$ is the $j$-th component of the mapping $F:M\to \mathbb{R}^n$ defining the integrable system. Denoting by $X_j\in\mathfrak{X}(M;\mathbb{R})$ the hamiltonian vector field associated to the $j$-th component of the mapping $F:M\to \mathbb{R}^n$, $f_j\in C^\infty(M;\mathbb{R})$, and $\exp(t_jX_j)\in\mathrm{Diff}(M)$ its flow at time $t_j\in\mathbb{R}$, the action $\rho:\mathbb{R}^n\to\mathrm{Diff}(M)$ is simply the composition of the flows, i.e. $\rho(t_1,\dots,t_n):=\exp(t_1X_1)\circ\cdots\circ\exp(t_nX_n)$.

Now let me comment on each part of Liouville theorem and give a sketch of the proof.

The hamiltonian vector fields generated by its components define an integrable (in the Sussmann sense) distribution of the tangent bundle whose leaves are generically lagrangian, with isotropic singular leaves.

By definition (or construction), the infinitesimal action is generated by the hamiltonian vector fields, $\rho_{*_e}(v_1),\dots,\rho_{*_e}(v_n)$, and they provide a basis for the tangent space of an orbit at any of its points, and $\omega(\rho_{*_e}(v_j),\rho_{*_e}(v_k))=\{f_j,f_k\}_\omega=0$; therefore, each orbit passing through $p\in M$ is an isotropic submanifold given by the connected components of the preimage by the momentum mapping of $F(p)\in\mathbb{R}^n$.

The connected components of the preimage of regular values (regular leaves) are homogeneous $\mathbb{R}^n$ spaces; they are diffeomorphic to $\mathbb{R}^{n-m}\times\mathbb{T}^m$.

The orbits of this action passing through a point $p\in M$ are diffeomorphic to the quotient of $\mathbb{R}^n$ by the stabiliser subgroup $G_p:=\{g\in\mathbb{R}^n \ ; \ \rho(g)(p)=p \}$.

It is not hard see that the (regular) orbits are diffeomorfic to $\mathbb{R}^{n-m}\times\mathbb{T}^m$, and this is implied by algebraic facts. Compact regular orbits are the famous Liouville tori.

The foliation is a lagrangian fibration in a neighbourhood of each regular leaf; it defines a fibre bundle with lagrangian fibres.

In an open set $V\subset M$ where each $p,q\in V$ satisfies $G_p\cong G_q$ one has a fibre bundle. This is just an application of Frobenius (or Sussmann) theorem.

In conclusion, integrable systems induce a foliation on the symplectic manifold whose leaves are generically lagrangian (with isotropic singular leaves) and diffeomorphic to $\mathbb{R}^{n-m}\times\mathbb{T}^m$, and near each regular leaf the foliation is a lagrangian fibration.

Thus, the missing piece from Liouville theorem is the symplectic linearisation of the hamiltonian action near a regular orbit.

There are coordinates on a local trivialisation of each lagrangian leaf in which $\omega$ is in Darboux form and the flows induced by each $f_j$ are linear.

One needs to prove a technical lemma ---Poincaré lemma for regular foliations--- before proving the linearisation of the hamiltonian action.

Lemma: Let $\alpha\in\Omega^k(M;\mathbb{R})$ be a given closed $k$-form, $\mathrm{d}\alpha=0$, whose restriction to an integrable distribution of constant rank $\mathcal{P}\subset\mathfrak{X}(M;\mathbb{R})$ vanishes. Then, for each trivialising neighbourhood $A\subset M$ of the regular foliation defined by $\mathcal{P}$, there exists a $\beta\in\Omega^{k-1}(A;\mathbb{R})$ such that $\beta$ vanishes when restricted to $\mathcal{P}$ and $\alpha=\mathrm{d}\beta$ on $A$.

A proof of this result can be obtained by using a homotopy operator... it is actually the same proof of the traditional Poincaré lemma.

Linearisation Theorem: The hamiltonian $\mathbb{R}^n$-action of an integrable system on $(M,\omega)$ can be symplectically linearised near each of its regular orbits

Proof: Near each regular orbit there exists a trivial fibre bundle structure with the momentum mapping as the projection. In this local trivialisation of this lagrangian fibration, the coordinates of the basis are given by the functions $f_1,\dots,f_n\in C^\infty(M;\mathbb{R})$ and the fibres are covered by coordinate functions $y_1,\dots,y_n\in C^\infty(A;\mathbb{R})$, with $A\cong (\mathbb{R}^{n-m}\times\mathbb{T}^m)\times\mathbb{R}^n$ and the $m$ functions $y_{n-m+1},\dots,y_n$ periodic with periods given by $a_p\in\mathbb{R}^m$.

Applying the Poincaré lemma for regular foliations to the symplectic form $\omega$ on the trivial fibre bundle near a regular orbit, with the integrable distribution $\mathcal{P}:=\langle \rho_{*_e}(v_1),\dots,\rho_{*_e}(v_n) \rangle_{C^\infty(M;\mathbb{R})}$, one has $\theta\in\Omega^1(A;\mathbb{R})$ satisfying $\omega\big{|}_A=\mathrm{d}\theta$ and $\theta\big{|}_{\mathcal{P}}=0$.

Since $\theta\big{|}_{\mathcal{P}}=0$, for each $X_j:=\rho_{*_e}(v_j)$ it holds $\imath_{X_j}\theta=0$, and because $\omega\big{|}_A=\mathrm{d}\theta$ one has $\imath_{X_j}(\mathrm{d}\theta)=-\mathrm{d} f_j$; Thus, the Lie derivative of $\theta$ with respect to $X_j$ is \begin{equation} \mathcal{L}_{X_j}(\theta)=\imath_{X_j}(\mathrm{d}\theta)+\mathrm{d} (\imath_{X_j}\theta)=-\mathrm{d} f_j \ . \end{equation}

The condition $\theta\big{|}_{\mathcal{P}}=0$ also implies that $\theta=-\sum_{k=1}^{n}\theta_k\mathrm{d} f_k$ and the previous equation reads \begin{eqnarray} -\mathrm{d} f_j=\mathcal{L}_{X_j}(\theta)&=&\mathcal{L}_{X_j}\left(-\sum_{k=1}^{n}\theta_k\mathrm{d} f_k\right)=-\sum_{k=1}^{n}\mathcal{L}_{X_j}(\theta_k\mathrm{d} f_k) \nonumber \\ &=&-\sum_{k=1}^{n}\left(X_j(\theta_k)\mathrm{d} f_k+\theta_k\mathcal{L}_{X_j}(\mathrm{d} f_k)\right) \nonumber \\ &=&-\sum_{k=1}^{n}\left(X_j(\theta_k)\mathrm{d} f_k+\theta_k\imath_{X_j}(\mathrm{d}\circ\mathrm{d} f_k)+\theta_k\mathrm{d} (\imath_{X_j}\mathrm{d} f_k)\right) \nonumber \\ &=&-\sum_{k=1}^{n}X_j(\theta_k)\mathrm{d} f_k \ , \end{eqnarray}yielding $X_j(\theta_k)=\delta_{jk}$.

The nondegeneracy of $\omega$ actually implies that $\mathrm{d}\theta_1,\dots,\mathrm{d}\theta_n$ are linearly independent over $A$: \begin{equation} \omega\big{|}_A=\sum_{k=1}^{n}\mathrm{d} f_k\wedge\mathrm{d}\theta_k \ . \end{equation}Thus, the mapping defined by $(f_1,\dots,f_n,y_1,\dots,y_n)\mapsto (f_1,\dots,f_n,\theta_1,\dots,\theta_n)$ is a diffeomorphism of $A$.

The theorem is proved by now: in the coordinates $(f_1,\dots,f_n,\theta_1,\dots,\theta_n)$ the symplectic form is just the Darboux form on $A$ and the hamiltonian action is linear, i.e. it is given by \begin{equation} (f_1,\dots,f_n,\theta_1,\dots,\theta_n)\mapsto (f_1,\dots,f_n,\theta_1+t_1,\dots,\theta_n+t_n) \ , \end{equation}where $(t_1,\dots,t_n)\in\mathbb{R}^n$, because $X_j(\theta_k)=\delta_{jk}$.

Q.E.D.

Last remark: due to results of Eliasson and Miranda, this theorem also holds true near each nondegenerate compact orbit. This generalisation is nontrivial by the simple reason that (in general) there is no Poincaré lemma for singular foliations, even in this particular case of a foliation coming from an integrable system with nondegenerate type of singularities.

Solution 2:

Since $F_1,\dots,F_n$ are assumed to be functionally linearly independent at each point of $M_f$, this is also true near it, and on a open subset $A\subset M$ (containing $M_f$) one can find coordinate functions that complete the set given by $F_1,\dots,F_n$; let me call those functions $\varphi_1,\dots,\varphi_n$.

In my previous answer I mentioned that compact regular orbits are diffeomorphic to tori $\mathbb{T}^n$ and that

The foliation is a lagrangian fibration in a neighbourhood of each regular leaf; it defines a fibre bundle with lagrangian fibres.

The coordinate functions $F_1,\dots,F_n,\varphi_1,\dots,\varphi_n$ are exactly the ones used to show this claim (cf. problem of page 279 of Arnold's book).

As it is pointed out by Arnold (by the end of page 279), in these coordinates the symplectic form (over $A$) is not necessarily written as $\sum_{j=1}^{n}\mathrm{d}F_j\wedge\mathrm{d}\varphi_j$.

He proceeds by defining (page 282) functions $I_1,\dots,I_n$ and assumes that instead of $F_1,\dots,F_n,\varphi_1,\dots,\varphi_n$, one can use $I_1,\dots,I_n,\varphi_1,\dots,\varphi_n$ as coordinate functions (page 283, which reads: $\mathrm{det}(\partial\mathbf{I}/\partial\mathbf{f})|_{\mathbf{f}}\neq 0$).

Remark: Whilst Arnold proceeds by changing the coordinate functions $F_j$, in my first answer I change the coordinate functions $\varphi_j$; that is the main difference in our arguments, and this is why I can provide a semilocal proof directly.

Let me show here how he constructs these new coordinate functions. His constructions assume that the symplectic manifold is $\mathbb{R}^{2n}$ endowed with the canonical symplectic structure, here I am not going to assume this: as the Original Poster is interested in the global picture.

A Liouville torus is defined as the preimage of a point $\mathbf{f}\in\mathbb{R}^n$ by the momentum mapping $(F_1,\dots,F_n):M\to\mathbb{R}^n$,

$$M_\mathbf{f}:=\{x\in M \ ; \ (F_1(x),\dots,F_n(x))=\mathbf{f}\} \ , $$

and choosing $[\gamma_1],\dots,[\gamma_n]\in H_1^\infty(M_\mathbb{f};\mathbb{Z})$ generators of the first smooth singular homology group of the torus $M_\mathbb{f}$ (for details about this terminology, I refer the readers to Lee's Introduction to smooth manifolds, but this is what Arnold is doing with his cycles by the end of page 282) one can define functions $I_j:A\subset M\to\mathbb{R}$ by

$$I_j(x):=\int_{\gamma_j}\theta \ ,$$

where the $1$-form $\theta\in\Omega^1(A;\mathbb{R})$ is any differential form satisfying $\omega=\mathrm{d}\theta$ over the open subset $A\subset M$ (cf. page 198 of Arnold's book, and my comments on the Poincaré lemma with parameters; otherwise, one can simply assume that $M\cong\mathbb{R}^{2n}$, as Arnold does in page 282), and $\gamma_j$ are smooth singular cycles representing $[\gamma_j]$. For points $x\in M_\mathbf{f}\cap A$ this is well defined (problem of page 283 in Arnold's book); when $x\in A$ does not belong to a Liouville torus which is a preimage of $\mathbb{f}$, the fact that $A\cong\mathbb{T}^n\times\mathbb{R}^n$ implies that it belongs to another Liouville torus and the cycles are to be taken from that particular torus.

Now I can discuss the theorem of page 283 in Arnold's book whose proof is the reason why I am writing this. It states that the coordinate functions $I_1,\dots,I_n,\varphi_1,\dots,\varphi_n$ satisfy

$$\omega=\sum_{j=1}^{n}\mathrm{d}I_j\wedge\mathrm{d}\varphi_j \ , $$

over the whole open subset $A\subset M$.

I hope that the reader can appreciate the fact that $A\subset M$ is a neighbourhood of the torus $M_\mathbf{f}$ and that these coordinates are well defined on the whole of it (which is exactly what the Original Poster asked).

Now, for the proof Arnold observes that any $\theta$ satisfying $\omega=\mathrm{d}\theta$ over $A\subset M$ is closed when restricted to the submanifold $M_\mathbf{f}$, as the restriction of $\omega$ vanishes there (the torus is a lagrangian submanifold). This observation allows him to define a function $S:V\subset M_\mathbf{f}\to\mathbb{R}$ satisfying $\theta=\mathrm{d}S$ over a contractible neighbourhood $V\subset M_\mathbf{f}$ of a fixed point $x_0\in M_\mathbf{f}$ (cf. page 198 of Arnold's book). This is related to my use of the foliated version of the Poincaré lemma in my first answer.

By the definition of the coordinate functions $I_j$, and applying Stokes's Theorem, it "holds"

$$I_j=\int_{\gamma_j}\theta=\int_{\gamma_j}\mathrm{d}S=\int_{\partial\gamma_j}S=\Delta_jS \ .$$

The quotation marks on "holds" is due to $S$ not being well defined on the whole of $M_\mathbf{f}$ (a fact observed by Arnold at the beginning of page 284).

He, then, uses Darboux coordinate functions $p_1,\dots,p_n,q_1,\dots,q_n$ over a contractible open neighbourhood $W\subset M$ of the point $x_0\in M_{\mathbf{f}}$ (one can take it in such a way that its intersection with $M_{\mathbf{f}}$ is $V$), and applies the Hamilton-Jacobi method that he has developed in chapter 9 to conclude the proof of the theorem; with the generating function being

$$S(I_1,\dots,I_n,q_1,\dots,q_n)=\int_{\mathbf{q}_0}^{\mathbf{q}}\theta \ , $$

where the integral is computed from any curve in $M_{\mathbf{f}}$ joining $\mathbf{q}_0:=(q_1(x_0),\dots,q_n(x_0))\in M_{\mathbf{f}}$ and $\mathbf{q}:=(q_1,\dots,q_n)\in M_{\mathbf{f}}$.

The local coordinates $p_1,\dots,p_n,q_1,\dots,q_n$ are chosen by exploiting the fact that $M_{\mathbf{f}}\subset M$ is a lagrangian submanifold (a null manifold as Arnold calls it in page 274); thus, a small piece of the torus, $V=W\cap M_{\mathbf{f}}$, can be realised as a graph of a function $\mathbf{p}:\mathbb{R}^n\to\mathbb{R}^{2n}$. This graph has the property that the values of the coordinate functions $I_1,\dots,I_n$ are constant along it, and they vary as one moves from one small piece of Liouville torus $V\subset M_{\mathbf{f}}$ to a small piece of a neighbouring Liouville torus (not the same value as $\mathbf{f}$). This is why $I_1,\dots,I_n,q_1,\dots,q_n$ can be taken as independent coordinate functions.

It is important to remark that the use of the Hamilton-Jacobi method (at least the way it is developed in the book) implies that the proof is local, it does not automatically extends to a neighbourhood of a Liouville torus. However, if $\omega=\sum_{j=1}^{n}\mathrm{d}I_j\wedge\mathrm{d}\varphi_j$ holds near every point of $M_\mathbf{f}$, then one can conclude that this is true over a neighbourhood of $M_\mathbf{f}$, where the coordinate functions $I_1,\dots,I_n,\varphi_1,\dots,\varphi_n$ are well defined.

P.D: A user asked me to not delete a previous version of this answer, but I could not keep my promise to do so. The ones who have upvoted this answer are welcome to follow the proof of the linearisation theorem on my first answer to the original question.

Arnold's theorem on action-angles.

Solution 1:

Solution 2:

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