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

$\displaystyle f: M \rightarrow \mathbb{R}^n, f = (f_1, \dots, f_n).$ (more…)

Let us consider a manifold ${M}$ and a hypersurface ${N \subset M}$. Given a smooth function ${F: T^*M \rightarrow \mathbb{R}}$, we are interested in solving for ${u \in C^{\infty}(M)}$ the eikonal equation

$\displaystyle \boxed{ F( du ) = 0 \ (*)}$

where the initial condition is ${u|_N = s}$ for ${s: N \rightarrow \mathbb{R}}$ smooth.

(The phrase “eikonal equation” seems to refer to a specific case of this, but I’ll follow Taylor in applying the term to this more general situation.)

Using the story already discussed about symplectic forms, we can do this locally in the noncharacteristic case.

To define this, we pick local coordinates ${x^1,\dots, x^n}$ around a fixed ${p_0 \in N}$ such that ${N}$ is given by ${x^n = 0}$. Let the corresponding local coordinates for ${T^*M}$ be ${x^1, \dots, x^n}$ together with ${\xi^1, \dots, \xi^n}$. Choose a cotangent vector ${v = \sum v_i dx^i \in T_{p_0}^*(M)}$ such that ${(ds)_{p_0} = \sum_{i < n} v_i dx^i}$. Suppose ${F(v)=0}$, and ${\frac{\partial F}{\partial \xi_n} \neq 0}$. This is the noncharacteristic hypothesis.

Theorem 1 Under the noncharacteristic hypothesis, the eikonal equation can be locally solved—that is, there exists a neighborhood ${U}$ containing ${p_0}$ and a smooth function ${u}$ on ${U}$ satisfying (*) with ${u|_{U \cap N} = s}$.  Moreover, $du|_{p_0} = v$.

The idea of the proof of this existence theorem is to construct ${u}$ by constructing ${du}$, which in turn is done by constructing its graph—which after all must satisfy a specific equation—as a submanifold of ${T^*M}$. At first it will not be obvious that the graph actually corresponds to any ${du}$. This will follow from the analysis of lagrangian submanifolds. (more…)

Yesterday I discussed the Poisson bracket on a symplectic manifold and some of its basic properties. Naturally enough, someone decided to axiomatize all this.

Poisson manifolds

So, a Poisson manifold is a smooth manifold ${M}$ together with a Lie algebra structure ${\{\cdot, \cdot\}}$ (the Poisson bracket) on the space of smooth functions on ${M}$ such that

$\displaystyle \{ fg, h \} = f\{g, h\} + g \{f, h \}.$

To check that a symplectic manifold is indeed a smooth manifold, we need only recall that there is a vector field ${H_h}$ with ${\{ f, h \} = -H_h f}$, so the above is just the derivation identity. All the other properties of the Poisson bracket on a symplectic manifold were established yesterday.

Now let’s switch to the general Poisson manifold case.

It turns out that the Poisson structure alone is enough to show many similarities. Indeed, the derivation identity above implies that there is a vector field, still denoted by ${H_h}$, with

$\displaystyle H_h f = - \{ f, h \}$

and in particular it follows that ${H_f f = 0}$ by antisymmetry. The set of such vector fields ${H_h}$ will be called, as in the symplectic case, Hamiltonian vector fields. I claim that the identity

$\displaystyle \boxed{ [H_f, H_g] = H_{ \{f,g\}} }$ (more…)

So, now that we have a notion of symplectic manifolds, we can talk about the Poisson bracket.   This gives a way of making the smooth functions into a Lie algebra.  The first step in the story is to use the symplectic form to associate to a function a vector field (obtained by duality from $df$).  These Hamiltonian vector fields have many nice properties: for instance, their Lie bracket is of the same type.  Moreover, they (and, locally, only they) are the vector fields whose flows preserve the symplectic form.  In mechanics, the flows of the Hamiltonian field associated to the energy function trace out the paths of a particle acted on by a conservative force.

Let ${M, \omega}$ be a symplectic manifold. Given a smooth function ${f: M \rightarrow \mathbb{R}}$, we have a 1-form ${df}$ on ${M}$. The self-duality of ${TM}$ induced by ${\omega}$ can be used to “lower indices” (kind of like how one gets a gradient on a Riemannian manifold) so that we get a vector field. Call its opposite ${H_f}$, the Hamiltonian vector field associated to ${f}$.

In other words,

$\displaystyle \sigma( H_f, V) = -df(V) = -Vf.$

By nondegeneracy, this uniquely determines ${H_f}$. (more…)

Now I’ll actually give the proof of the Darboux theorem that a symplectic manifold is locally symplectomorphic to $\mathbb{R}^{2n}$ with the usual form.

Proof of the Darboux theorem

We will prove the equivalent:

Theorem 1 Let ${M}$ be a manifold with closed symplectic forms ${\omega_0, \omega_1}$, and ${p \in M}$ with ${\omega_0(p) = \omega_1(p)}$. Then there are neighborhoods ${U,V}$ of ${p}$ and a diffeomorphism ${f: U \rightarrow V}$ with ${f^*\omega_1 = \omega}$.

The idea is to consider the continuously varying family of 2-forms

$\displaystyle \omega_t = (1-t) \omega_0 + t \omega_1 = \omega_0 + t \alpha$

where

$\displaystyle \alpha = \omega_1 - \omega_0 .$

We will consider a small neighborhood ${U}$ of ${p}$ and a smooth map ${ G: U \times [0,1] \rightarrow M }$ such that ${G_t:=G(\cdot, t)}$ is a diffeomorphism, ${G_0 = id}$, and

$\displaystyle G_t^*(\omega_t) = \omega_0 \ (*).$ (more…)

The purpose of this post is to discuss a few basic facts about differentiable manifolds and state the Darboux theorem, which I will prove next time.  (People who are looking for a more ambitious leap into symplectic geometry might want to try lewallen’s two posts over at Concrete Nonsense.)

A symplectic manifold is a smooth manifold ${M}$ equipped with a closed symplectic 2-form ${\omega}$. In other words, ${\omega}$ is alternating and nondegenerate on each tangent space ${T_p(M)}$

The basic example of a symplectic form is

$\displaystyle \sum_i dx^i \wedge d\xi^i$

on ${\mathbb{R}^{2n}}$ with coordinates ${x^i, \xi^i, 1 \leq i \leq n}$

This can also be written in a more invariant form, which will also give an invariant manner of making the cotangent bundle ${T^*M}$ of any manifold ${M}$ into a symplectic manifold. First, we define a 1-form ${\alpha}$ on ${T^*M}$. Let ${p: T^*M \rightarrow M}$ be the projection downwards. Given ${v \in T^*M}$ lying above ${\xi \in T^*M}$, define

$\displaystyle \alpha(v) = \xi( p_*(v)).$

To make this clearer, here is an interpretation in local coordinates. Let ${x^1, \dots, x^n}$ be local coordinates for ${M}$. Then ${x^1, \dots, x^n, \xi^1, \dots, \xi^n}$ coordinates for ${T^*M}$. Then

$\displaystyle \alpha = \sum \xi^i dx^i$

as is easily checked by working through the definitions. So we can define a canonical 2-form ${\omega}$ as ${\omega = - d \alpha}$; this makes ${T^*M}$ into a symplectic manifold. (more…)