Compactification of \(\mathbb{R}^ k\)-actions and Hamiltonian systems of toric type. (Compactification d’actions de \(\mathbb{R}^ k\) et systèmes hamiltoniens de type torique.) (French) Zbl 0810.58015

Veeravalli, Alain

C. R. Acad. Sci., Paris, Sér. I 317, No. 3, 289-292 (1993).

The author’s summary: “Let’s consider a Hamiltonian system \((M, \omega, H)\) endowed with a finite dimensional space \(\mathcal A\) of first integrals. The center of \(\mathcal A\) for the Poisson brackets defines an infinitesimal Hamiltonian action of \(R^ k\) on \(M\). If \(F\) is a compact orbit of this action, we show that a reasonable hypothesis on the function jets gives a normal form of “toric type” in a neighbourhood of \(F\). This result, conjectured by P. Molino [Sémin. Gaston Darboux, Géom. Topologie Différ. 1989-1990, No. 5, 39-47 (1991; Zbl 0734.70013)], generalizes the theorems of Arnold-Liouville, Eliasson, Nekhoroshev and its version with singularities of Dufour. The key of the proof is an “\(R\)-action compactification” theorem which extends a result of J. P. Dufour and P. Molino”.

Reviewer: D.Nguyen Huu (Dalat)

MSC:

37J99

Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems

Keywords:

Hamiltonian systems; toric type; symplectic manifold; desingularization