Action Angle Coordinates and Symplectically complete istropic foliation

An old result by V. I. Arnold states that if M is symplectic of dimension 2n and f:M–> P is a submersion with Lagrangian fibers (in particular P has dimension n), then locally around a point x in P with compact fiber, M admits action angle coordinates. Namely, coordinates of the form (a_i, alpha_j): i, j ranges over 1 to n, such that the alpha_i curves correspond to a_i hamiltonian flows, i.e., d/d(alpha_i) = H_{a_i}; f only depends on a_i’s, and furthermore alpha_i’s take value in S^1 (so really they are like angles), or R/2pi Z if you will. The proof is very much in the same spirit as the flow proof of Darboux’s theorem in symplectic geometry, which was given by Weinstein. The original proof by Arnold uses some technique called quadrature, which does the job by integration of 1-forms (namely d of the angle coordinates constructed bofore the action ones) along loops.
Now J. J. Duistermaat generalized that result to global setting, i.e., finding admissible criterion for global action angle coordinates to exist on such a pair M –>P, assuming M, P compact and fibers are all compact. There turn out to be two simple topological invariants associated with this Lagrangian submersion, namely the Chern class associated with the quasi-fiber-bundle M–>P, and the monodromy associated with the covering R–>P, where R is the period lattice in T*P of the Hamiltonian flows. In the following I will define some terms and show how they are relevant to the problem at hand.

So at each point x of P, we can associated artificially an n-dimensional vector space corresponding to the different direction of hamiltonian flows we can take at any point in the preimage of x, with H_{a_i} a standard basis. The local AA (action angle) theorem enables us to write out bundle charts for this disjoint union of vector spaces, hence they form a legit vector bundle over P. Call it B–>P. Using the fact that the fibers of M–> P are lagrangian, we can show there is a unique 1-form eta_i whose pullback by f:M–>P is equal to da_i for each i, where a_i recall are the LOCAL action coordinates. So why do we bother with this? Because it allows us to identify that artificial vector bundle over P with its cotangent bundle T*P, which can be done simply by identifying H_{a_i} with da_i and extend by linearity.

Now let R be a multiple valued section of T*P –> P, given by the periodic points of the n-dimensional hamiltonian flow in the artificial bundle B, which can immediately be identified with a section of T*P–>P. So basically for each vector in a fiber of B at some point x in P, we can think of it as representing a recipe for how to flow along the Hamiltonian vector field, which of course depends on which starting point we take in the fiber of x in M –> P. Now R is simply those vectors that represent a flow which brings any (hence all) point in the fiber of x back to itself. Clearly R is discrete over each x, since only Z combination of H_{a_i} gives isotropic flows. Bear with me that R is a Z^n covering space over P imbedded in T*P through the correspondence of B and T*P. We can take the quotient T*P/R, which makes a fiber bundle over P. The important thing now is that this fiber bundle has a natural fibered action on M–>P, namely the action given by the Hamiltonian flow bundle B, identified with T*P, and passed down in a well-defined manner to the quotient T*P/R. This is a well-defined action simply because R is the core (le noyau) of the fibered action of T*P on M –>P, which at each fiber is simply the isotropy of the action restricted to that fiber.

Now what this action enables us to do is to examine the extent to which M–>P is isomorphic to the fiber bundle T*P/R –> P. Here is how: if we take a finite atlas of P, say U_i with contractible intersections U_i cap U_j. Then over each intersection, we can define a section sigma_ij of T*P/R–>P which when acted on the the intersection viewed as a subset of U_i gives the correct adjustment between their local angle variables, so that the angle variables associated with U_i and with U_j match. Now the sigma_ij together form a 1-cocycle (being the difference of smooth sections of 0-chains) in the sense of Cech cohomology. Its differential lies in the so-called Chern class associated with the fiber bundle M–>P, which lies in the second sheaf cohomology group of germs of lagrangian sections of P in T*P or T*P/R (lagrangian because the angle coordinates over each U_i are given by lagrangian sections as their starting points). Using the fact that sheaf of lagrangian sections is a subsheaf of smooth sections, which is a fine sheaf (i.e., partition of unity exists), the initial segment of the cohomology sequence splits into two pieces, one of which shows that the degree 1 cohomology group is isomorphic to degree 2 cohomology group via the Cech differential. So the chern class gives exactly the same amount of information as the 1-cocycle sigma_ij, up to 1-coboundaries. But coboundary is irrelevant in defining the bundle M–>P, so chern class determines the bundle. It turns out that T*P/R is isomorphic to M–>P precisely when the chern class vanishes.

Now the second obstruction to realizing global action angle coordinates is the so-called monodromy associated with the covering R–>P. Basically that’s just the covering group based at any point in P, which is a subgroup of GL(n, Z), if we identify preimage in R of any point x in P with Z^n. The result by Duistermaat says in the case Chern class vanishes, M admit global action angle coordinates precisely when the monodromy group at any (hence all) point in P is the trivial group. It’s easy to see that triviality of monodromy is necessary, but the converse is certainly not obvious, and remarkable.

The next step is to generalize Duistermaat’s result to the case when dimension of P is lower than n. But actually a bit more is needed about the actual coordinate functions of the submersion M–>P. It turns out that a fruitful direction is to consider k functions on M (k <n+1)in involution with not only each other, but also an independent set of coordinate functions containing those k functions on M. And let P be the image of those k functions, as ordered k-tuple. This was taken up by a russian mathematician named Nehorosev who concluded that instead of action angle variables on M, we get something a little more complicated, but still quite reminiscent to the original model, namely we have four sets of local coordinates, I, phi, p, and q, with phi being a variable mod 2pi, or taking value in S^1 if you like. Such that the symplectic form on M is locally of the form d(I_i dphi_i + p_j dq_j). Note that when dimension of P is n, the p’s and q’s are gone, so we are back to the Arnold scenario. For the global question, Nehorosev didn’t give a necessary and sufficient condition. But he did prove some nontrivial results in that direction.

The next stage is even more general and concerns with the setting of a foliation F on the symplectic manifold M. (I will explain the basics about foliation theory in a different thread) It’s more general because a submersion always gives a regular foliation on the source manifold, whose leaves are simply the connected component of the fibers. Furthermore we require that the foliation is istropic, i.e., the distribution (i.e., vector subbundle of the tangent bundle) it defines is contained in its symplectic orthogonal complement; or in symbols, F subset F^{omega}, where omega is the symplectic form on M. Also we require the foliation to be symplectically complete, meaning that its symplectic orthogonal distribution also forms a regular foliation. One sees immediately that Nehorosev’s setting is a special case of this, since the condition that the k coordinate functions on M Poisson commute with 2n-k functions containing themselves means precisely that the foliation is coisotropic, meaning its symplectic orthogonal contains the distribution it defines. Furthermore since that 2n-k dimensional foliation defined by the k coordinate functions clearly yields a regular foliation, we see that it’s symplectically complete and we can reverse the role of the original foliation and its symplectic complement and get back to the isotropic situation. So much for the generalization, now let’s state some results.

So given such an FISC (Feuilletage Isotrope Symplectiquement Complet) F of dimension k, locally we can work in a foliation chart, in which we have the natural local submersion M–>P, where P is the space of plaques of the foliation, with an inherited Poisson structure of rank 2r=2n-2k. It is then clear from Nehorosev’s result that locally M is symplectomorphic to R^(2r) x T*T^k, where T^k is the k-dimensional torus. One can similarly consider the Chern class and monodromy group etc and obtain nice results about globalizability of such local symplectomorphism. Unfortunately my French is not so premium and what I read hardly remains in my memory, without excessive repetition. So here is the main reference: Pierre Dazord & Thomas Delzant: Le Probleme General Des Variables Actions-Angles, J. Differential Geometry 26 (1987) 223-251

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About aquazorcarson

math PhD at Stanford, studying probability
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