KAM I

In this post I want to sketch the idea of KAM, following these lecture notes.

Integrable Systems

I don't want to worry too much about details, so for now we'll define an integrable system to be a Hamiltonian system $(M, \omega, H)$ for which we can choose local Darboux coordinates $(I, \phi)$ with $I \in \mathbb{R}^N$ and $\phi \in T^N$, such that the Hamiltonian is a function of $I$ only. Defining $\omega_j := \partial H / \partial I_j$, Hamilton's equations then read
$$\begin{align} \dot{I}_j &= 0, \\\ \dot{\phi}_j &= \omega_j(I). \end{align}$$
Hence we obtain linear motion on the torus as our dynamics. Note in particular that the sets $\{I = \mathrm{const}\}$ are tori, and that the dynamics are constrained to these tori. We call these tori "invariant".


Now suppose that our Hamiltonian $H$ is of the form
$$H(I, \phi) = h(I) + f(I, \phi)$$
with $f$ "small". What can be said of the dynamics? Specifically, do there exist invariant tori? KAM theory lets us formulate this question in a precise way, and gives an explicit quantitative answer (as long as $f$ is nice enough, and small enough).

I want to sketch the idea of the KAM theorem, completely ignoring analytical details.

Constructing the Symplectomorphism

Suppose we could find a symplectomorphism $\Phi$: (I, \phi) \mapsto (\tilde{I}, \tilde{\phi})\) such that $H(I, \phi) = H(\tilde{I}$. Then our system would still be integrable (just in new action-angle coordinates), and we'd be done. There are two relatively easy ways of constructing symplectomorphisms: integrating symplectic vector fields, and generating functions. In the lecture notes, generating functions are used, so let's take a minute to discuss them.

Proposition Let $\Sigma(\tilde{I}, \phi)$ be a smooth function and suppose that the transformation
$$I = \frac{\partial \Sigma}{\partial \phi}, \ \tilde{\phi} = \frac{\partial \Sigma}{\partial \tilde{I}}$$
can be inverted to produce a diffeomorphism $\Phi: (I, \phi) \mapsto (\tilde{I}, \tilde{\phi})$. Then $\Phi$ is a symplectomorphism.

Proof
$$dI = \frac{\partial^2 \Sigma}{\partial \phi \partial \tilde{I}} d \tilde{I}$$
$$d\tilde{\phi} = \frac{\partial^2 \Sigma}{\partial \phi \partial \tilde{I}} d\phi$$
Hence
$$dI \wedge d\phi = \frac{\partial^2 \Sigma}{\partial \phi \partial \tilde{I}} d \tilde{I} \wedge d\phi = d\tilde{I} \wedge d\tilde{\phi}.$$

We want a symplectomorphism $\Phi$ such that
$$H \circ \Phi(\tilde{I}, \tilde{\phi}) = \tilde{h}(\tilde{I}$$
If $\Phi$ came from a generating function $\Sigma$, then we have
$$H(\frac{\partial \Sigma}{\partial \phi}, \phi) = \tilde{h}(\tilde{I})$$
Expanding things, we have
$$h(\frac{\partial \Sigma}{\partial \phi}) + f(\frac{\partial \Sigma}{\partial \phi}, \phi) = \tilde{h}(\tilde{I}).$$

If $f$ is small, then we might expect $\Phi$ to be close to the identity, and hence $\Sigma$ ought to be close to the generating function for the identity (which is $\langle I, \phi \rangle$). So we take
$$\Sigma(\tilde{I}, \phi) = \langle \tilde{I}, \phi \rangle + S(\tilde{I}, \phi)$$
where $S$ should be "small". So we linearize the equation in $S$:
$$\langle \omega(\tilde{I}), \frac{\partial S}{\partial \phi} \rangle + f(\tilde{I}, \phi) = \tilde{h}(\tilde{I}) - h(\tilde{I})$$

Now we can expand $S$ and $f$ in Fourier series and solve coefficient-wise. This gives a formal solution $S(\tilde{I}, \phi)$ of the equation
$$\langle \omega, \frac{\partial S}{\partial \phi} \rangle + f(\tilde{I}, \phi) = 0.$$

Getting it to Work

Unfortunately, the Fourier series for $S$ has no chance to converge, so instead we take a finite truncation. If we assume $f$ is analytic, its Fourier coefficients decay exponentially fast, so this provides a very good approximate solution to the linearized equation (and we can give an explicit bound in terms of a certain norm of $f$). Call this function $S_1$. We then use $S_1$ to construct a symplectomorphism $\Phi_1$.

Now we take
$$H_1(I, \phi) = H \circ \Phi_1(I, \phi) = h_1(I) + f_1(I, \phi).$$
Some hard analysis then shows that $h - h_1$ is small, and $f_1$ is much smaller than f.

The Induction Step

The above arguments sketch a method to put the system "closer" to an integrable form. By carefully controlling $\epsilon$'s and $\delta$'s, one then shows that iterated sequence $\Phi_1, \Phi_2 \circ \Phi_1, \ldots$ converges to some limiting symplectomorphism $\Phi_\infty$.