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Compactification of actions of \(\mathbb{R}^ n\) and action-angle variables with singularities. (Compactification d’actions de \(\mathbb{R}{} ^ n\) et variables action-angle avec singularités.) (French) Zbl 0752.58011
Symplectic geometry, groupoids, and integrable systems, Sémin. Sud- Rhodan. Geom. VI, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 20, 151-167 (1991).
[For the entire collection see Zbl 0722.00026.]
The authors consider an infinitesimal action of \(\mathbb{R}^ 2\) on a symplectic manifold \(V\) which is generated by a vector space of first integrals (or Hamiltonian functions). They show that if the singularities of these functions are “of elliptic type” then, near any compact orbit, there is an action of the torus \(T^{2n}\) which has the same orbits and commutes with the given action of \(\mathbb{R}^{2n}\). As a corollary, they recover the Eliasson’s theorem on the existence of singular action-angle variables for a Hamiltonian system.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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.)