Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics

Izosimov, Anton; Khesin, Boris; Mousavi, Mehdi

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.

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

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