## May 12, 2015

### 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, $\mathcal{M}$.