Sketch proof of the Arnold-Liouville theorem

This is a high-level, informal summary of the proof of the Arnold-Liouville theorem as stated and proved in V. I. Arnold's Mathematical Methods of Classical Mechanics (chapter 10, page 271). Due acknowledgement must also be given to Maciej Dunajski's notes on Integrable Systems (web), which I used as a reference. Any errors below are mine.

$$ \def\real{{\mathbb{R}}} \def\q{{\mathbf{q}}} \def\p{{\mathbf{p}}} \def\Phi{{\boldsymbol{\phi}}} \def\I{{\mathbf{I}}} \def\f{{\mathbf{f}}} \def\c{{\mathbf{c}}} \def\t{{\mathbf{t}}} $$

Let $M = \real^{2n}$ be a phase space in $n$ degrees of freedom, and let $H(\q, \p)$ be a Hamiltonian. Let $\f = (f_k(\q, \p))_k$ be $n$ functions that are independent (so $\nabla f_k$ linearly independent) and in involution (so $\{f_j, f_k\} = 0$), with $f_1 = H$. Then this system is said to be integrable in the Arnold-Liouville sense.

For a given vector $\c = (c_k)$ denote the level sets of $\f$ as $$M_{\f=\c} = \{(\q, \p)\in M | \f(\q, \p) = \c\}.$$ Since the $f_k$ are first integrals, the trajectories $(\q(t), \p(t))$ takes place on $M_{\f=\c}$ for fixed values of $\f = \c$ given by the initial conditions.

The Arnold-Liouville theorem makes a statement about the topology of these manifolds, and constructs a new canonical coordinate system $(\Phi, \I)$ of "action-angle coordinates", through a sequence of finitely many algebraic operations and integrals of known functions. In practice, these operations may be nontrivial or require the introduction of special functions.

In these new coordinates the equations of motion take a particularly simple form, such that the full solution can be completely determined from an initial condition.

Theorem (Arnold-Liouville). Let $(M, H, \f)$ form an integrable system as above. Then:

1. If $M_{\f=\c}$ is compact and connected, then it is diffeomorphic to a torus $\mathbb{T}^n = S^1 \times \dots \times S^1$.
2. In the neighbourhood of $M_{\f=\c}$, there is a canonical coordinate system $(\Phi, \I)$ such that the $I_k$ are first integrals.
3. The equations of motion become of the form $\dot{\phi_k} = \omega_k(I)$, $\dot{I_k} = 0$, for known functions $\omega_k$: so the $I_k$ are first integrals and the $\phi_k$ can be found by integrations of known functions.

Jump to part1, part2, part3.

Part 1: Topology of $M_{\f=\c}$

Hamiltonian vector fields and Hamiltonian flows

By the assumption that the $f_k$ are independent, each level set $M_{\f=\c}$ is an $n$-dimensional manifold, with the values of $c_k = f_k(\q(0), \p(0))$ determined by the initial conditions.

For a given function $h(\q, \p)$ we introduce the Hamiltonian vector field – an operator – corresponding to $h$ defined by $$X_h = \{h, \cdot\} = \sum_{a, b}\omega^{ab} \frac{\partial h}{\partial \xi^a}\frac{\partial}{\partial \xi^b}.$$ Recall that if $h = H$ is the Hamiltonian then for any other function $f$ $$ \frac{\mathrm d f}{\mathrm d t} = \{h, f\}. $$ Thus the Hamiltonian vector field $X_h$ generates the flows that would arise if the function $h$ were a Hamiltonian. Specifically, $X_h$ generates the group $$ g_h^t: M \rightarrow M $$ for $t \in \mathbb{R}$, with $g_h^0 = \mathrm{id}$. The action is "take $x_0 = (\q_0, p_0)$ as an initial condition, and time-evolve by taking $h$ to be a Hamiltonian".

The first integrals generate commuting Hamiltonian flows

(page 273 in Arnold)

Let $X_{f_k}$ be the Hamiltonian vector field of the first integral $f_k$; and for the two functions $f_j$ and $f_k$ write the corresponding flows as $g_j^{t_j}$ and $g_k^{t_k}$.

The involution condition$\{f_j, f_k\} = 0$ implies that $$[X_{f_j}, X_{f_k}] = 0$$ and so the flows that they generate also commute: $$ g_j^{t_j} g_k^{t_k} x_0 = g_k^{t_k} g_j^{t_j} x_0 \quad \forall x_0\in M. $$

Let $\t = (t_k)$ be a vector of times and consider the group action $$ g^\t = g_{1}^{t_1} \dots g_{n}^{t_n} $$ on $M$. "Time-evolve along each the flow generated by $f_k$ for time $t_k$." Everything commutes so $g^{\t + \t'} = g^{\t}g^{\t'}$, with $g^\mathbf{0} = \mathrm{id}$. Thus we have a group action of $\real^n$ on $M$.

Periodic dynamics on the level sets

(page 274 in Arnold)

The involution condition $\{f_j, f_k\} = 0$ implies that each $f_k$ is conserved under the flow created by each $f_j$. In particular the flows $g^\t$ preserve the level sets $M_{\f=\c}$.

Choose an arbitrary point $\mathbf{x} \in M_{\f=\c}$; then the "time-evolution" map $\real^n \rightarrow M_{\f=\c}$ is defined by $$\t \mapsto g^\t \mathbf{x}.$$

Now assume that the level set $M_{\f=\c}$ is compact and connected.

Properties of the time-evolution map:

• Surjective: For any $\mathbf{y} \in M_{\f=\c}$ there is some $\t$ such that $g^\t \mathbf{x} = \mathbf{y}$. Follows from connectedness and the fact that the $f_k$ form a coordinate system on the level set.
• Identity: $g^\mathbf{0} \mathbf{x} = \mathbf{x}$.
• Non-injective: $\real^n$ is not compact, so there must be some "period vector" $\mathbf{T}$ such that $g^\mathbf{T} \mathbf{x} = \mathbf{x}$.
• Injective near the origin: There is $\epsilon$ such that for $|\t| < \epsilon$, we have $g^\t \mathbf{x} \neq \mathbf{x}$ except for $\t = \mathbf{0}$.

Level sets are toruses

(page 276 in Arnold)

Surjectivity implies that the set of periods $\mathbf{T}$ for which $g^\mathbf{T} \mathbf{x} = \mathbf{x}$ is in fact independent of the choice of $\mathbf{x}$. Dimensional arguments imply that there are $n$ linearly independent period vectors. Injectivity near the origin implies that these $\mathbf{T}$ form a lattice $\mathbb{Z}^n$ (they cannot be arbitrarily close to the origin) and so in fact $$M_{\f=\c} \cong \real^n / \mathbb{Z}^n \cong \mathbb{T}^n.$$

Part 2: Constructing the action-angle coordinates

(page 281 in Arnold)

The goal now is to create a canonical coordinate transformation $(\q, \p) \mapsto (\Phi, \I)$ such that the action coordinates $\I$ are first integrals. In general it is not acceptable to use the original first integrals $\f$ as these do not necessarily give rise to a canonical coordinate transformation.

Assume that the system $\f(\q, \p) = \c$ can be inverted for $\p = \p(\q, \c)$, and so $\f(\q, \p(\q, \c)) = \c$ identically.

We shall do this by setting up a generating function $S(\q, \I)$ such that $$ \p = \frac{\partial S}{\partial \q}, \quad \Phi = \frac{\partial S}{\partial \I}. $$ Thus $$ dS |_{\I} = \p \cdot d\q $$ along paths of constant $\I$.

Noting that $\I$ is constant on level sets, define $$ S = \int_{\q_0}^\q \p \cdot d\q $$ between points $\q_0$ and $\q$ on the torus. It can be shown, by Stokes' theorem, that this integral does not change under continuous deformations of the path of integration: in other words, that $dS = \p \cdot d\q$ is an exact differential. However, $S$ is multivalued in $\q$ since nonzero contributions can come from "winding the contour around the torus".

Thus, let $\gamma_k$ be the $n$ cycles on the torus, arbitrarily chosen up to continuous deformation, and define $$ I_k = \frac{1}{2\pi} \oint _{\gamma_k} \p \cdot d\q $$

as the action variables. These are well-defined since they do not depend on any particular choice of cycle, and are intrinsic to the manifold $M_{\f=\c}$; they therefore are functions $\I(\f)$ of the original functions $\f$.

Having defined the generating function $S$ and the action variables $\I$ it now follows to define the angle coordinates by $$ \Phi = \frac{\partial S}{\partial \I}. $$ These define a position on the torus and each is periodic with period $2\pi$ – adding $2\pi$ to an angle coordinate corresponds to taking an integration contour that wraps around the torus an extra time.

Part 3: The new equations of motion

Since the coordinate transformation was produced by a generating function $S$, it is canonical.

Write $\tilde{H}(\I) = H(\q, \p)$ for the Hamiltonian now as a function of the action variables (it cannot depend on $\Phi$ since those identify a position on the torus, while $H$ is constant on the torus). Then the equations of motion simply become $$ \dot{\phi_k} = \omega_k := \partial \tilde{H}/\partial I_k, \quad \dot{I_k} = 0 $$ noting that the frequencies $\omega_k$ are also constant. Therefore given initial conditions we then have $\I(t) = \I(0)$ and $\phi_k(t) = \phi_k(0) + \omega_k(t)$.