On the linearization of Hamiltonian systems on Poisson manifolds (Q2508691)

Let \((M,\{.,.\},H)\) be a Hamilton-Poisson system. Acording to the general scheme, the linearization procedure applied to the dynamical system \((M,X_H)\) defines a vector field Var\((X_H)\) on the tangent bundle \(TM\). Let \((N,\omega)\) be a closed symplectic leaf of \((M,\{.,.\})\), \(T_{N}M\) the restriction of the tangent bundle \(TM\) to the leaf \(N\) and Var\(_{B}(X_H)\) the restriction of Var\((X_H)\) to the normal bundle \(T_{B}M/TB\) of \(B\). It is clear that Var\(_{B}(X_H)\) is not in general an Hamiltonian vector field, in fact the linearization destroy the Hamiltonian property. In the paper under review the author formulates some results on the existence of an Hamiltonian structure for Var\(_{B}(X_H)\). A nice example is also pointed out.