Today I’ll say a few words on completely integrable systems.  There is a nice result of Liouville-Arnold that describes how these are fibered, at least when the fibers are compact.  It will provide a useful illustration of the ideas discussed already.

Charles Siegel of Rigorous Trivialties has a post on this topic at a much more sophisticated level.  There was also a successful MO question by Gil Kalai about them.

A completely integrable system is a symplectic manifold ${M}$ (with symplectic form ${\omega}$) of dimension ${2n}$ together with ${n}$ smooth functions ${f_1, \dots, f_n}$ in involution, i.e. $\displaystyle \{f_i, f_j \} = 0$

for all ${i,j}$, and with the differentials ${df_i}$ independent at each cotangent space ${T_x^*(M)}$. Physically, this corresponds to a set of conservation laws. For instance, if ${f_1}$ is energy, then a particle goes through the integral curves of ${H_{f_1}}$. Thus because ${H_{f_1} f_i = 0}$, ${f_i}$ is constant on these integral curves.

Now, fix a point ${c \in \mathbb{R}^n}$. We consider the map $\displaystyle f: M \rightarrow \mathbb{R}^n, f = (f_1, \dots, f_n).$

It is a regular map by the assumption about the ${df_i}$, so ${M_c := f^{-1}(c)}$ is a smooth ${n}$-dimensional submanifold for each ${c \in \mathbb{R}^n}$.

I claim it is actually a lagrangian submanifold. Indeed, first note that the flows of the ${H_{f_i}}$ preserve each ${M_c}$ (cf. the above remarks on conservation laws). It follows that at each point ${x \in f^{-1}(c)}$, $\displaystyle \{ H_{f_i}(x) , 1 \leq i \leq n\} \subset T_x(M_c),$

and these are linearly independent tangent vectors, because they correspond via the nondegneerate symplectic form ${\omega}$ to the linearly independent differentials.

So in particular, the ${H_{f_i}(x)}$ form a basis for each such tangent space. Since ${\omega(H_{f_i}, H_{f_j}) = \{f_i,f_j\} \equiv 0}$, we get the lagrangian result.

Theorem 1 (Liouville-Arnold) If ${M_c}$ is compact and connected, then it is diffeomorphic to ${T^n = \mathbb{R}^n/\mathbb{Z}^n}$, the ${n}$-dimensional torus.

The idea is to define a morphism ${\mathbb{R}^n \rightarrow M_c}$ using the Hamiltonian flows. So pick ${x \in M_c}$, and consider the flows ${\phi_i( \cdot, \cdot)}$ which correspond to ${H_{f_i}}$. Since ${H_{f_i}}$ can be viewed as a vector field on the manifold ${M_c}$, compactness implies these flows are globally defined. Moreover since $\displaystyle [H_{f_i}, H_{f_j}] = H_{\{f_i, f_j\}} = H_0=0$

the vector fields, hence the flows, commute. So define $\displaystyle g(t_1, \dots, t_n,x) = \phi_1(\phi_2(\dots \phi_n(x, t_n), t_{n-1}), \dots, t_n)).$

Since the flows commute, it follows that $\displaystyle g(u,g(t,x))=g(u+t,x) \ \mathrm{ for} \ u,t \in \mathbb{R}^n. \ (*)$

Set ${g_x := g(\cdot, x)}$. I claim that ${g_x}$ is locally a diffeomorphism, and it will yield the diffeomorphism to the torus when quotiented out appropriately. By the identity (*), we just have to prove this at ${t=0}$ (with ${x}$ arbitrary). Now then the image of the differential contains each ${H_{f_i}(x)}$ because of the use of the flows, and these tangent vectors generate, so we have a submersion. Since the dimensions are equal, this means we have a local diffeomorphism.

Now consider ${S \subset \mathbb{R}^n}$ consisting of points ${s}$ with ${g_x(s)=x}$. Then (*) implies ${S}$ is a subgroup of ${\mathbb{R}^n}$, which is discrete since we have the local diffeomorphism condition. The quotient group ${\mathbb{R}^n/S}$ is thus mapped (glboally) diffeomorphically onto ${M_x}$, again in view of (*). Since this is compact, ${S}$ must be a lattice in ${\mathbb{R}^n}$ of maximal rank, thus isomorphic to ${\mathbb{Z}^n}$.

There is also a fact that the flows of these Hamiltonian fields can actually be constructed by quadrature, i.e. using basic operations of integration, differentiation, algebraic manipulation, and taking inverses.  For the proof, cf. Taylor’s book (Partial Differential Equations, Basic Theory).