Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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 (,ω)(\mathcal{M},\omega) be a 2n2n-dimensional symplectic manifold. Let us posit that \mathcal{M} had a foliation by nn-dimensional Lagrangian tori (a torus, TMT\subset M, is Lagrangian if ω| T=0\omega|_T =0). Removing a subset, SS\subset \mathcal{M}, of codimension codim(S)2codim(S)\geq 2, where the leaves are singular, we can assume that all of the leaves on =\S\mathcal{M}'=\mathcal{M}\backslash S are smooth tori of dimension nn.

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

  1. The φ i\varphi^i restrict to angular coordinates on the tori. In particular φ i\varphi^i shifts by 2π2\pi when you go around the corresponding cycle on TT.
  2. The K iK_i are globally-defined functions on \mathcal{M} which are constant on each torus.
  3. The symplectic form ω=dK idφ i\omega= d K_i\wedge d \varphi^i.

From 1, it’s clear that it’s more convenient to work with the 1-forms dφ id\varphi^i, which are single-valued (and closed, but not necessarily exact), rather than with the φ i\varphi^i themselves. In 2, it’s rather important that the K iK_i are really globally-defined. In particular, an integrable Hamiltonian is a function H(K)H(K). The K iK_i are the nn 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, =T *M\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}.

Posted by distler at 11:49 AM | Permalink | Followups (13)