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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 (ℳ,ω)(\mathcal{M},\omega) be a 2n2n-dimensional symplectic manifold. Let us posit that ℳ\mathcal{M} had a foliation by nn-dimensional Lagrangian tori (a torus, T⊂MT\subset M, is Lagrangian if ω| T=0\omega|_T =0). Removing a subset, S⊂ℳS\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 ℳ′=ℳ\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 ℳ\mathcal{M} which are constant on each torus.
  3. The symplectic form ω=dK i∧dφ 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, ℳ=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, ℳ\mathcal{M}.

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Posted by distler at 11:49 AM | Permalink | Followups (26)