completely integrable system

Informal overview

Integrable systems have applications in mechanics. Say, for a physical system, you are given a symplectic manifold (that usually means some type of phase space) and a Hamiltonian function .

The goal is usually to understand the “trajectories” of the system, meaning the integral curves of the Hamiltonian vector field (lets call it ), even better is to understand ALL the integral curves and get a picture of the flow of . Unfortunately, this is usually very hard.

In a special case, if enough conserved quantities can be identified, the flow (as a bunch of differential equations) becomes easier to solve (AKA integrate). Note that the Hamiltonian function itself is a conserved quantity by design. is constant on integral curves of . So, if we can find other functions that are also constant along integral curves of , that are “different” enough functions (here’s where the functional independence condition in the second part of the definition comes in), then we can find a chart that preserves the symplectic form that is MUCH simpler to solve. This comes with fancy tools like generating functions and action-angle coordinates. See @abraham2008 chapter 5.2 for more about the physics.

Definition

On symplectic manifold

Let be a symplectic manifold of dimension . A completely integrable system is a smooth map

where each pairwise Poisson commute, i.e.

and is a smooth submersion almost everywhere. (In other words, the set of regular values is an open, dense subset.)

On Poisson manifold

The definition for a Poisson manifold is almost the same, but instead we need Poisson commuting, functionally independent functions.

As a maximal Poisson commuting subalgebra

In both the symplectic and Poisson commuting case, we want to define a maximal Poisson-commuting subalgebra of .

Symplectic case

Given a symplectic manifold with , there can be at most Poisson commuting functions that are functionally independent.

Proof

Suppose there are smooth functions such that .

If we pick any point , we can look at the vector subspace defined as

Where denotes the Hamiltonian vector field for the function .

Looking at the symplectic form on this subspace,

Thus, by linearity of the symplectic form, vanishes, so is an isotropic subspace.

This means that .

Next, assume that the functions are independent. Thus, is surjective (almost everywhere).

Note that for the standard basis of ,

Using linearity of the differential, we have

Written somewhat more clearly,

So this can only be surjective if is a linearly independent set of .

Since is non-degenerate, we can look at the isomorphism

Note that since are Hamiltonian vector fields,

So thus, the subspace , and there can be at most independent functions.

Poisson manifold case

Given a Poisson manifold , there can be at most Poisson commuting functions that are functionally independent.

Proof

By the Weinstein splitting theorem, we can pick small enough open sets centered at each point such that

where has the symplectic form induced from .

Since is an open set from a symplectic manifold there can only be independent functions that Poisson commute on . is not symplectic, so we can look for independent functions that Poisson commute.

Note that we can always include independent Casimir functions, since they Poisson commute with every function, so they will automatically commute with the other functions already found. There can only be other independent functions since .

Thus, there can be functions (for the largest dimension symplectic leaf).

Cool things

• There is a relationship between certain Hamiltonian torus actions and completely integrable systems shown here.