Posted by: lewallen | February 26, 2011

Harmonic oscillators, integrable systems, and the Arnold-Liouville theorem

Last weekend, the Princeton math department hosted a retreat on integrable systems. About 30 people came, and half of us prepared talks. The goal was to teach each other some of the background to the (large and amorphous) field of integrable systems, with a focus (as far as it went) on the KdV equation, Hitchin’s system, and spectral curves. I have to say, it was an absolutely great experience, and I highly recommend other grad students organizing such things, if possible. Actually, we had such a good time that a few of us are planning a similar “retreat” (this time, on the Princeton campus) in May on mirror symmetry and related things. If you’re in the area and interested in giving a talk, let me know!

The talk I gave was an intro talk on integrable systems and the Arnold-Liouville theorem. It’s funny, when I first wrote about symplectic geometry here, more than a year ago, I didn’t understand it at all — everyone in the comments was telling me to pay more attention to the physics, and I wanted to, but I had only ever heard of symplectic things from the topologist’s point of view, and I had a hard time wrapping my head around this other perspective. Now, after the retreat, I’m a complete convert and I think everyone should learn the physics (and dynamics!) first.

But for this post, I’ll assume some knowledge of the basic set-up, namely symplectic manifolds and Hamiltonian vector fields. We start with a symplectic manifold (M,\omega) and a Hamiltonian H: M\to \mathbb{R}. This can be thought of as measuring the energy of our system. We’re interested in the integral curves of the Hamiltonian vector field X_H (which we should think of as the trajectories of our system), and flowing along these curves will preserve H. If we can find another conserved quantity H_1: M\to \mathbb{R}, that is to say, another function preserved by the flow of X_H, this is called an integral for the system (this is equivalent to the vector fields X_H and X_{H_1} commuting). If we can find n linearly independent mutually commuting integrals, where 2n is the dimesion of M, our system is called fully integrable, or integrable in the sense of Arnold-Liouville. It turns out that these systems are beautifully structured and symmetric (this is a precise manifestation of Noether’s principle, that conserved quantities correspond to symmetries). Note, however, that the Arnold-Liouville theorem can almost never be used to explicitly solve the system. It is an existence theorem, although a powerful one: it says that one can (in theory) find (very well-behaved!) local coordinates in which the system looks just like a linear flow on a torus (so in particular, it’s recurrent and shows no “chaos”).

A beautiful and highly non-trivial example of this is the geodesic flow on a Euclidean ellipsoid, which I hope to get to later. For now, we start with the prototypical example, the 1 dimensional Harmonic oscillator (which models the behavior of a mass on a spring). The AL theorem can be thought of as saying that every fully integrable system locally looks like n copies of the Harmonic oscillator.

The phase space is

{M=\{(q,p)\}={\mathbb R}^{2}=T^{*}{\mathbb R}}.

Here, q measures the distance from the origin of the mass of our spring, and p=dq/dt is its momentum, with mass set to 1. The symplectic form is

\omega = dq \wedge dp = -d\lambda, ~\lambda = pdq

As our Hamiltonian we take the sum of the kinetic and potential energy of our system:

\displaystyle H = 1/2(p^{2} + q^{2})

The canonical equations (which tell us that our curve (p(t),q(t)) \subset M integrates the Hamiltonian vector field) are then

\displaystyle dq/dt = \partial H /\partial p = p

\displaystyle dp/dt = -\partial H / \partial q = -q

It’s easy to see that these are equivalent to the formulation

d^2 q/d^2t = -q

which we derive from Newton and Hooke’s laws.

We can solve these equations by inspection with

\displaystyle (q(t), p(t)) = (R\cos t, R \sin t)

(for a fixed {R\in {\mathbb R}}, and with initial conditions {(q(0),p(0))= (R,0)}). The solutions are a family of circles foliating the phase space, with a degenerate circle at the origin which is just a fixed point, and the unique critical point of {H}.

We can see this even better in polar coordinates, {(r,\theta)}. In these coordinates the new symplectic form is {1/r * dr\wedge d\theta}, and the canonical equations become

\displaystyle dr/dt = 0

\displaystyle d\theta/dt = 1

and therefore we can directly (via integration!) solve them and get

\displaystyle (r(t),\theta(t)) = (R,t)

as expected. So polar coordinates seem perfectly made for this problem. In fact, we could actually take our coordinate to be {\mathcal{R} = 1/2 r^{2}}, then the form is just {d\mathcal{R}\wedge d \theta}. Note that {\mathcal{R}} is exactly our Hamiltonian! Therefore the first equation just expresses the fact that the Hamiltonian is conserved, and the second expresses the fact that the {\theta} circles are exactly the Hamiltonian orbits. Therefore we get a foliation of our phase space by Hamiltonian orbits, parameterized by the values of the Hamiltonian. In fact it’s a principal foliation, with the circle group acting on the orbits via the flow of {H}. The {\theta} coordinate is just given by the integral curves of the flow.

2. One dimension up

The previous case was a bit trivial, because the Hamiltonian {H} was the only integral of motion that we needed to get full integrability. Let’s quickly inspect the 2-dimensional case and see how it’s an exact generalization of the 1d case. Here we have {M = \{x=(q_{1},p_{1},q_{2},p_{2})\} = {\mathbb R}^{4}}. The form is

\displaystyle \omega = \sum_{i}dq_{i}\wedge dp_{i}=-d\lambda , ~\lambda = \sum p_{i}dq_{i}

The hamiltonian is

\displaystyle H = 1/2 \sum_{i}(p_{i}^{2}+a_{i}q^{2}_{i})

and our equations are

\displaystyle dq_{i}/dt = p_{i}

\displaystyle dp_{i}/dt = -q_{i}

Again we can solve this via sines and cosines. Note that things might be a bit more complicated because if the {a_{i}} are independent over {{\mathbb Q}}, there won’t be many closed oribts! Still, we can check that the radial functions

\displaystyle \mathcal{R}_{i} = 1/2(p_{i}^{2}+q^{2}_{i})

are still 2 linearly independent integrals of motion. So as before we change coordinates to polar: {(\mathcal{R}_{1},\theta_{1},\mathcal{R}_{2},\theta_{2})} If we take a mutual level set {F= \{\mathcal{R}_{i}= c_{i}\}} of the two integrals, then if neither is 0, the level set {(\theta_{1},\theta_{2})=S^{1}\times S^{1}} is just a torus, parameterized by the flows of closed Hamiltonian orbits of {\mathcal{R}_{1}} and {\mathcal{R}_{2}}, just as before. Because these are integrals, the level sets are also level sets for {H}, and we can see that the restricted canonical equations near this torus take the form

\displaystyle d\mathcal{R}_{i}/dt = 0

\displaystyle d\theta_{i}/dt = a_{i}

and in these coordinates our solution is

\displaystyle (R_{1},a_{1} t ,R_{2},a_{2}t)

So it’s just a linear combination of linear things on the torus level sets, although it might be an irrational linear combination (so the orbits aren’t periodic). Again, our space foliates into principal tori. Note that there are a few circles and points, the degenerate level sets, where one or both of the {\mathcal{R}_{i}} have critical points.

3. Upshot

What have we seen? Take our system {M} with integrals {H_{1},\dots, H_{n}}. Put them together into a function {\mathcal{H}=(H_{1},\dots, H_{n}): M\rightarrow {\mathbb R}^{n}}. The level sets of \mathcal{H}, on which the Hamiltonian H is constant, are tori, which are naturally parameterized by the integral curves {\{\theta_{i}(t)\}} of the Hamiltonian orbits of the {H_{i}}, and locally, the coordinates {\{ H_{i},\theta_{i}\}} give a foliation into principal tori. Near these tori the equations look like {d H_{i}/dt = 0, d \theta_{i}/dt= a_{i}} for some constants, and so restricted to these tori the flow looks linear.

Said differently, we define a “Hamiltonian torus action:”

Definition: Suppose we have a torus action \rho: T^n \to Symp(M) (so the torus acts by diffeomorphisms which preserve the symplectic form). If we choose a basis X_1, \dots, X_n for the Lie algebra of T^n, \rho gives us n corresponding vector fields on M, generating the action. If these vector fields are Hamiltonian we call the action a Hamiltonian torus action. Letting H_1,\dots, H_n be a choice of n corresponding Hamiltonians, we call \mathcal{H} = (H_1,\dots ,H_n): M\to \mathbb{R}^n the moment map for the action.

Then, the Arnold-Liouville theorem says that, at least in the special case where the level sets of all the integrals are compact, a Hamiltonian system is fully integrable if and only if it admits local Hamiltonian torus actions (non-degenerate away from the isolated critical points of the integrals), on whose orbits the Hamiltonian H is constant. In fact, the coordinates of the moment map \mathcal{H} give the n integrals. M is therefore locally foliated by tori, with coordinates {\{ H_{i},\theta_{i}\}}, in which the canonical equations restricted to the leaves of the foliation is just a linear system on a torus.

Some remarks/corollaries:

  • the torus leaves or fibers are lagrangian. in fact, the form locally looks like {\sum_{i}dH_{i}\wedge d\theta_{i}}.

    And, we can choose or integrals appropriately so that:

  • The natural linear (circle) coordinates {\theta_{i}} are just the integral curves of the hamiltonian fields {X_{H_{i}}}. These are called angle coordinates.
  • The integrals {\{H_{i}\}} which provide a complementary set of coordinates, can be obtained simply by integrating a local primitive {\lambda} for the symplectic form {\omega} along the closed cycles for the coordinates {\theta_{i}}:
    \displaystyle H_{i}(v) = \int_{\theta_{i}.v} \lambda (Y_{\theta_{i}}.v) (ie, they are cycles of the canonical 1-form). The Hamiltonian {\mathcal{H}} is just a function of these action coordinates {\mathcal{H}=\mathcal{H}(H_{1},\dots H_{n})}.
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