### Action-Angle Variables

This semester, I taught the Graduate Mechanics course. As is often the case, teaching a subject leads you to rethink that you *thought* you understood, sometimes with surprising results.

The subject for today’s homily is *Action-Angle* variables.

Let $(\mathcal{M},\omega)$ be a $2n$-dimensional symplectic manifold. Let us posit that $\mathcal{M}$ had a foliation by $n$-dimensional
*Lagrangian* tori (a torus, $T\subset M$, is Lagrangian if $\omega|_T =0$). Removing a subset, $S\subset \mathcal{M}$, of codimension $codim(S)\geq 2$, where the leaves are singular, we can assume that all of the leaves on $\mathcal{M}'=\mathcal{M}\backslash S$ are
*smooth* tori of dimension $n$.

The objective is to construct coordinates $\varphi^i, K_i$ with the following properties.

- The $\varphi^i$ restrict to angular coordinates on the tori. In particular $\varphi^i$ shifts by $2\pi$ when you go around the corresponding cycle on $T$.
- The $K_i$ are globally-defined functions on $\mathcal{M}$ which are
*constant*on each torus. - The symplectic form $\omega= d K_i\wedge d \varphi^i$.

From 1, it’s clear that it’s more convenient to work with the 1-forms $d\varphi^i$, which are single-valued (and closed, but not necessarily exact), rather than with the $\varphi^i$ themselves. In 2, it’s rather important that the $K_i$ are really
*globally*-defined. In particular, an *integrable Hamiltonian* is a function $H(K)$. The $K_i$ are the $n$ conserved quantities which make the Hamiltonian integrable.

Obviously, a given foliation is compatible with infinitely many “integrable Hamiltonians,” so the existence of a foliation is the more fundamental concept.

All of this is totally standard.

What never really occurred to me is that the standard construction of action-angle variables turns out to be very closely wedded to the particular case of a cotangent bundle, $\mathcal{M}=T^*M$.

As far as I can tell, action-angle variables don’t even *exist* for foliations of more general symplectic manifolds, $\mathcal{M}$.

^{1}, as we move between tori of the foliation. The $K_i$ are then defined as

Surely, there’s some sort of cohomological characterization of when Action-Angle variables exist. The situation feels a lot like the characterization of when symplectomorphisms (vector fields that preserve the symplectic form) are actually
*Hamiltonian* vector fields^{2}.

#### Update:

Just to be clear, there are plenty of examples where you*can*construct action-angle variables for foliations of symplectic manifolds which are not cotangent bundles. An easy example is $(\mathcal{M},\omega) = \left(S^2, \frac{r dr\wedge d\theta}{{(1+r^2)}^2}\right)$ where $(K,\varphi)= \left(-\frac{1}{2(1+r^2)},\theta\right)$ are action-angle variables for the obvious foliation by circles. This example “works” because once you remove the singular leaves (at $r=0,\infty$), $\omega$ becomes cohomologically trivial on $\mathcal{M}'$ and we can then use the standard construction. $[\omega]=0\in H^2(\mathcal{M}\backslash S,\mathbb{R})$ sounds like a sufficient condition for constructing action-angle variables. But is it necessary?

^{1}I’m pretty sure we need them to be globally-constant over $\mathcal{M}'$. I’ll assume there’s no obstruction to doing that.

^{2}If you’re not familiar with that story, note that
$\mathcal{L}_X \omega = 0$
is tantamount to the condition that $i_X\omega$ is a closed 1-form. If it happens that it is an *exact* 1-form,
$i_X\omega = d f$
then $X = \{f,\cdot\}$ is a Hamiltonian vector field. The obstruction to writing $X$ as a Hamiltonian vector field is, thus, the de Rham cohomology class, $[i_X\omega]\in H^1(\mathcal{M},\mathbb{R})$.

In the example at hand, that’s exactly what is going on. Any single-valued function, $H(\theta_1,\theta_2)$, is an “integrable” Hamiltonian for the above foliation. But the symmetries, $X_1=\frac{\partial}{\partial\theta_3}$ and $X_2=\frac{\partial}{\partial\theta_4}$ are *not* Hamiltonian vector fields. Hence, there are no corresponding conserved action variables.

## Re: Action-Angle Variables

Which textbook did you use?