Generating Functions

Method of Generating Functions

Let \(X\) and \(Y\) be two smooth manifolds, and let \(M = T^\ast X, N = T^\ast Y\) with corresponding symplectic forms \(\omega_M\) and \(\omega_N\).

Question: How can we produce symplectomorphisms \(\phi: M \to N\)?

The most important construction from classical mechanics is the method of generating functions. I will outline this method, shameless stolen from Ana Cannas da Silva's lecture notes.

Suppose we have a smooth function \(f \in C^\infty(X \times Y)\). Then its graph \(\Gamma\) is a submanifold of \(M \times N\): \( \Gamma = \{ (x,y, df_{x,y}) \in M \times N \}\). Since \(M \times N\) is a product, we have projections \(\pi_M, \pi_N\), and this allows us to write the graph as
\[ \Gamma = \{ (x, y, df_x, df_y) \}\]
Now there is a not-so-obvious trick: we consider the twisted graph \(\Gamma^\sigma\) given by
\[ \Gamma^\sigma =  \{(x,y, df_x, -df_y) \} \]
Note the minus sign.

Proposition If \(\Gamma^\sigma\) is the graph of a diffeomorphism \(\phi: M \to N\), then \(\phi\) is a symplectomorphism.

Proof By construction, \(\Gamma^\sigma\) is a Lagrangian submanifold of \(M \times N\) with respect to the twisted symplectic form \(\pi_M^\ast \omega_M - \pi_N^\ast \omega_N\). It is a standard fact that a diffeomorphism is a symplectomorphism iff its graph is Lagrangian with respect to the twisted symplectic form, so we're done.

Now we have:

Modified question: Given \(f \in C^\infty(M \times N)\), when is its graph the graph of a diffeomorphism \(\phi: M \to N\)?

Pick coordinates \(x\) on \(X\) and \(y\) on \(Y\), with corresponding momenta \(\xi\) and \(\eta\). Then if \(\phi(x,\xi) = (y,\eta)\), we obtain
\[ \xi = d_x f, \ \eta = -d_y f \]
Note the simlarity to Hamilton's equations. By the implicit function theorem, we can construct a (local) diffeomorphism \(\phi\) as long as \(f\) is sufficiently non-degenerate.

Different Types of Generating Functions

We now concentrate on the special case of \(M = T^\ast \mathbb{R} = \mathbb{R} \times \mathbb{R}^\ast\). Note that this is a cotangent bundle in two ways: \(T^\ast \mathbb{R} \cong T^\ast \mathbb{R}^\ast\). Hence we can construct local diffeomorphisms \(T^\ast \mathbb{R} \to T^\ast \mathbb{R}\) in four ways, by taking functions of the forms

\[ f(x_1, x_2), \ f(x_1, p_2), \ f(p_1, x_2), \ f(p_1, p_2) \]

Origins from the Action Principle, and Hamilton-Jacobi

Suppose that we have two actions

\[ S_1 = \int p_1 \dot{q}_1 - H_1 dt, \ S_2 = \int p_2 \dot{q}_2 - H_2 dt \]

which give rise to the same dynamics. Then the Lagrangians must differ by a total derivative, i.e.

\[ p_1 \dot{q}_1 - H_1 = p_2 \dot{q}_2 - H_2  + \frac{d f}{dt} \]

Suppose that \(f = -q_2 p_2 + g(q_1, p_2, t)\). Then we have

\[ p_1 \dot{q}_1 - H_1 = -q_2 \dot{p}_2 - H_2 + \frac{\partial g}{\partial t} + \frac{\partial g}{\partial q_1}\dot{q}_1 + \frac{\partial g}{\partial p_2} \dot{p_2} \]

Comparing coefficients, we find
\[ p_1 = \frac{\partial g}{\partial q_1}, \ q_2 = \frac{\partial g}{\partial p_2}, \ H_2 = H_1 + \frac{\partial g}{\partial t} \]

Now suppose that the coordinates \((q_2, p_2)\) are chosen so that Hamilton's equations become
\[ \dot{q_2} = 0, \ \dot{p}_2 = 0 \]
Then we must have \(H_2 = 0\), i.e.
\[ H_1 + \frac{\partial g}{\partial t} = 0 \]
Now we also have \(\partial H_2 / \partial p_2 = 0\), so this tells us that \(g\) is independent of \(p_2\), i.e. \(g = g(q_1, t)\). Since \(p_1 = \partial g / \partial q_1\), we obtain
\[ \frac{\partial g}{\partial t} + H_1(q_1, \frac{\partial g}{\partial q_1}) = 0 \]
This is the Hamilton-Jacobi equation, usually written as
\[ \frac{\partial S}{\partial t} + H(x, \frac{\partial S}{\partial x}) = 0 \]
Note the similarity to the Schrodinger equation! In fact, one can derive the Hamilton-Jacobi equation from the Schrodinger equation by taking a wavefunction of the form
\[ \psi(x,t) = A(x,t) \exp({\frac{i}{\hbar} S(x,t)}) \]
and expanding in powers of \(\hbar\). This also helps to motivate the path integral formulation of quantum theory.

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 \(\Phi_\infty\).

Circle Diffeomorphisms I

This is the first of a series of posts based on these lecture notes on KAM theory. For now I just want to outline section 2, which is a toy model of KAM thoery.

Circle Diffeomorphisms

We consider a map \(\phi: \mathbb{R} \to \mathbb{R}\) defined by
\[ \phi(x) = x + \rho + \eta(x) \]
where \(\rho\) is its rotation number and \(\eta(x)\) is "small".

Define \(S_\sigma\) to be the strip \(\{ |\mathrm{Im} z|<\sigma\} \subset \mathbb{C}\) and let \(B_\sigma\) be the space of holomorphic functions bounded on \(S_\sigma\) with sup norm \(\|\cdot\|_\sigma\).

Goal: Show that if \(\|\eta\|_\sigma\) is sufficiently small, then there exists some diffeomorphism \(H(x)\) such that
\[ H^{-1} \circ \phi \circ H (x) = x + \rho \]
i.e. that \(\phi\) is conjugate to a pure rotation.

Linearization

The idea is that if \(\eta\) is small, then \(H\) should be close to the identity, so we suppose that
\[ H(x) = x + h(x) \]
where \(h(x)\) is small. Plugging this into the equation above and discarding higher order terms yields
\[ h(x+\rho) - h(x) = \eta(x) \]
Since \(\eta\) is periodic, we Fourier transform both sides to obtain an explicit formula for the Fourier coefficients of \(h(x)\). We have to show several things:

1. The Fourier series defining \(h(x)\) converges in some appropriate sense.

2. The function \(H(x) = x + h(x)\) is a diffeomorphism.

3. The composition \(\tilde{\phi} = H^{-1} \circ \phi \circ H\) is closer to a pure rotation than \(\phi\), in the sense that
\[ \tilde{\phi}(x) = x + \rho + \tilde{\eta}(x) \]
where \(\|\tilde{\eta}\| \ll \|\eta\|\).

Newton's Method

Carrying out the analysis, one finds that for appropriate epsilons and deltas, if \(\eta \in B_\sigma\) then \(H \in B_{\sigma - \delta}\) and that \(\|\tilde{\eta}\|_{\sigma-\delta} \leq C \|\eta\|_\sigma^2\). By carefully choosing the deltas, we can iterate this procedure (composing the \(H\)'s) to obtain a well-defined limit \(H_\infty \in B_{\sigma/2}\) such that
\[ H_\infty^{-1} \circ \phi \circ H_\infty (x) = x + \rho, \]
as desired.

So in fact the idea of the proof is extremely simple, and all of the hard work is in proving some estimates.