×

zbMATH — the first resource for mathematics

Géometrie globale des systèmes hamiltoniens complètement intégrables: Fibrations lagrangiennes singulières et coordonnées action-angle à singularités. (Global geometry of completely integrable Hamiltonian systems: Lagrangian singular fibrations and action-angle variables with singularities). (French) Zbl 0702.58032
Initially, the observation is made that the study of the global geometry of a completely integrable hamiltonian system devolves upon the study of triples (M,\(\omega\),\({\mathcal F})\), where (M,\(\omega\)) is a symplectic manifold and \({\mathcal F}\) is a (Stefan-Sussmann) singular foliation on M subject to certain involution, compactness and elliptic transversality assumptions. The “characteristic triad” (W,\({\mathcal R},\gamma)\) of such a triple is then defined as being composed of the space of leaves \(W=M/{\mathcal F}\), a naturally constructed lattice \({\mathcal R}\) in the cotangent bundle of W, and the Chern class \(\gamma\) of the lagrangian fibration \(\pi\) : \(M\to W\). The space W is a locally convex flat affine manifold with affine boundary and corners, endowed with a boundary- adapted affine atlas in which coordinate changes are affine transformations whose linear parts have integral coefficients. Such a manifold will be called a BM-manifold in this review.
A realization theorem for characteristic triads is then stated. Let W be a BM-manifold, \({\mathcal R}\) a lattice defined as before, and \(\gamma\) an element of \(H^ 2(W,{\mathcal R})\) lying in the kernel of a canonical map of that group into \(H^ 3(W,{\mathbb{R}})\). Then there exist a symplectic manifold (M,\(\omega\)) and a singular foliation \({\mathcal E}\) on M (satisfying the assumptions referred to above) such that the characteristic triad of (M,\(\omega\),\({\mathcal F})\) is (W,\({\mathcal R},\gamma)\); and the triple (M,\(\omega\),\({\mathcal F})\) is unique up to a diffeomorphism of M and a closed 2-form on W. That theorem generalizes results due to J. J. Duistermaat, On global action-angle coordinates, Commun. Pure Appl. Math. 33, 687-706 (1980; Zbl 0439.58014), and to T. Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. Fr. 116, No.3, 315-339 (1988; Zbl 0676.58029)].
A necessary and sufficient condition for the existence of global action- angle coordinates with singularities, adapted to a triple (M,\(\omega\),\({\mathcal F})\) as above, is announced at the end. The criterion is that \({\mathcal R}\) be trivial, \(\gamma\) be zero and \(\omega\) be exact. In this connection cf. e.g. L. H. Eliasson, Normal forms for hamiltonian systems with Poisson commuting integrals - elliptic case, Comment. Math. Helv. 65, No.1, 4-35 (1990; Zbl 0702.58024); J.-P. Francoise, Intégrales de périodes en géométries symplectique et isochore, in Géométrie symplectique et mécanique, Lect. Notes Math. 1416, 105-138 (1990; Zbl 0702.58022); and P. Dazord, Groupoïdes symplectiques et troisième théorème de Lie “non linéaire”, ibid., 39-74 (1990; Zbl 0702.58023). See also the following review.
[Reviewer’s remarks: The French word “fibrations” in the title has been rendered into English as “foliations”. In this regard cf. e.g. p. 56 of the paper by P. Dazord mentioned above. In addition, the translation of the “Résumé” contains several errors.]
Reviewer: Hugo H.Torriani

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
55R55 Fiberings with singularities in algebraic topology
57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
53C70 Direct methods (\(G\)-spaces of Busemann, etc.)
PDF BibTeX XML Cite