Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
[Orbites coadjointes des difféomorphismes symplectiques des surfaces et hydrodynamique idéale]
Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2385-2433.

Nous présentons une classification des orbites coadjointes génériques pour les groupes de symplectomorphismes et de difféomorphismes hamiltoniens des surfaces fermées symplectiques. Nous classons également les fonctions de Morse simples sur les surfaces symplectiques par rapport à l’action de ces groupes. Cela donne une réponse au problème posé par V. Arnold sur la description des invariants de champs isorotationnels génériques dans des liquides idéaux en deux dimensions. Nous introduisons la notion de primitive sur un graphe de Reeb mesuré et nous décrivons ses propriétés.

We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V. Arnold’s problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3066
Classification : 58E40, 76M60, 58B25
Keywords: coadjoint orbits, symplectic diffeomorphisms, Hamiltonian diffeomorphisms, Casimirs, simple Morse functions, isovorticed fields, measured Reeb graphs, pants decomposition, vorticity function, circulations
Mot clés : orbite coadjointe, difféomorphisme symplectique, difféomorphisme hamiltonien, fonction de Casimir, fonction de Morse simple, champs isorotationnelles, graphe de Reeb mesuré, décompositions en pantalons, fonction de la vorticité, circulation
Izosimov, Anton 1 ; Khesin, Boris 1 ; Mousavi, Mehdi 2

1 Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 (Canada)
2 Department of Statistical and Actuarial Sciences, University of Western Ontario, London, ON N6A 5B7 (Canada)
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Izosimov, Anton; Khesin, Boris; Mousavi, Mehdi. Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics. Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2385-2433. doi : 10.5802/aif.3066. http://www.numdam.org/articles/10.5802/aif.3066/

[1] Arnold, V.I. On the representation of functions of several variables as a superposition of functions of a smaller number of variables, Mat. Prosveschenie, Volume 3 (1958, English transl.: V. Arnold, Collected Works, Volume 1, 25-46, 2009), pp. 41-61

[2] Arnold, V.I. Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’institut Fourier, Volume 16 (1966) no. 1, pp. 319-361 | DOI

[3] Arnold, V.I. Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, 1978, x+462 pages

[4] Arnold, V.I.; Khesin, B.A. Topological methods in hydrodynamics, Applied Mathematical Sciences, 125, Springer, New York, 1998, xv+374 pages

[5] Banyaga, A. Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Commentarii Mathematici Helvetici, Volume 53 (1978) no. 1, pp. 174-227 | DOI

[6] Bolsinov, A.V. A smooth trajectory classification of integrable Hamiltonian systems with two degrees of freedom, Sbornik: Mathematics, Volume 186 (1995) no. 1, pp. 1-27 | DOI

[7] Bolsinov, A.V.; Fomenko, A.T. Integrable Hamiltonian systems: geometry, topology, classification, CRC Press, 2004, xv+730 pages

[8] Bolsinov, A.V.; Oshemkov, A.A. Singularities of integrable Hamiltonian systems, Topological methods in the theory of integrable systems, Cambridge Scientific Publishers, 2006, pp. 1-67

[9] Choffrut, A.; Šverák, V. Local structure of the set of steady-state solutions to the 2D incompressible Euler equations, Geometric and Functional Analysis, Volume 22 (2012) no. 1, pp. 136-201 | DOI

[10] Dufour, J.-P.; Molino, P.; Toulet, A. Classification des systèmes intégrables en dimension 2 et invariants des modèles de Fomenko, Comptes rendus de l’Académie des sciences. Série 1, Mathématique, Volume 318 (1994) no. 10, pp. 949-952

[11] Earle, C.J.; Eells, J. A fibre bundle description of Teichmüller theory, Journal of Differential Geometry, Volume 3 (1969) no. 1-2, pp. 19-43

[12] Farb, B.; Margalit, D. A primer on mapping class groups, Princeton Mathematical Series, Princeton University Press, 2012, xiv+492 pages

[13] Gramain, A. Le type d’homotopie du groupe des difféomorphismes d’une surface compacte, Annales scientifiques de l’École Normale Supérieure (série 4), Volume 6 (1973) no. 1, pp. 53-66

[14] Hatcher, A.; Thurston, W. A presentation for the mapping class group of a closed orientable surface, Topology, Volume 19 (1980) no. 3, pp. 221-237 | DOI

[15] Khesin, B.; Misiołek, G. Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics, Volume 176 (2003) no. 1, pp. 116-144 | DOI

[16] Kirillov, A.A. Orbits of the group of diffeomorphisms of a circle and local Lie superalgebras, Functional Analysis and Its Applications, Volume 15 (1981) no. 2, pp. 135-137 | DOI

[17] Kirillov, A.A. The orbit method, II: Infinite-dimensional Lie groups and Lie algebras, Contemporary Mathematics, Volume 145 (1993), p. 33-33 | DOI

[18] Kruglikov, B.S. Exact smooth classification of Hamiltonian vector fields on two-dimensional manifolds, Mathematical Notes, Volume 61 (1997) no. 2, pp. 146-163 | DOI

[19] Maslov, V. P.; Shafarevich, A. I. Asymptotic solutions of Navier-Stokes equations and topological invariants of vector fields and Liouville foliations, Theoretical and Mathematical Physics, Volume 180 (2014) no. 2, pp. 967-982 | DOI

[20] Moser, J. On the volume elements on a manifold, Transactions of the American Mathematical Society, Volume 120 (1965) no. 2, pp. 286-294 | DOI

[21] Putman, A. A note on the connectivity of certain complexes associated to surfaces, L’Enseignement Mathématique, Volume 54 (2008), pp. 286-301

[22] Segal, G. Unitary representations of some infinite dimensional groups, Communications in Mathematical Physics, Volume 80 (1981) no. 3, pp. 301-342 | DOI

[23] Shnirelman, A.I. Lattice theory and flows of ideal incompressible fluid, Russian Journal of Mathematical Physics, Volume 1 (1993) no. 1, pp. 105-113

[24] Toulet, A. Classification des systèmes intégrables en dimension 2, Université de Montpellier 2, France (1996) (Ph. D. Thesis)

[25] Colin de Verdière, Y.; Vey, J. Le lemme de Morse isochore, Topology, Volume 18 (1979) no. 4, pp. 283-293 | DOI

[26] Witten, E. Coadjoint orbits of the Virasoro group, Communications in Mathematical Physics, Volume 114 (1988) no. 1, pp. 1-53 | DOI

[27] Wolf, U. Die Aktion der Abbildungsklassengruppe auf dem Hosenkomplex, Karlsruhe, Germany (2009) (Ph. D. Thesis)

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