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Stabilité et linéarisation dans les variétés de Poisson. (Stability and linearization in Poisson manifolds). (French) Zbl 0677.58024

Géométrie symplectique et mécanique, Rencontre Balaruc/France 1983, Sémin. sud-rhodanien Géom., 59-75 (1985).
[For the entire collection see Zbl 0606.00019.]
We know that every Poisson manifold (W,\(\Lambda)\) with Poisson tensor \(\Lambda\) [A. Lichnerowicz, J. Differ. Geom. 12, 253-300 (1977; Zbl 0405.53024)] is partitioned into weakly embedded symplectic submanifolds constituting a foliation with singularities. Here the author examines the specific properties of the characteristic foliation.
If V is a symplectically complete foliation and if the space W of its leaves is a manifold, then W is equipped canonically with a Poisson structure (locally that is always the case). This result, when applied to the study of Hamiltonian group actions in the sense of J.-M. Souriau [Structure des systèmes dynamiques (1970; Zbl 0186.580)] and to the study of stationary motions, generalizes previous results.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
58A30 Vector distributions (subbundles of the tangent bundles)