On Lie-Poisson systems
Let \(Q\) be a Lie group and \(K \subset Q\) a subgroup acting on \(T^* Q\) as a symmetry group for a Hamiltonian function \(H\) on \(T^* Q\). The representation \(T^* Q \to Q \times {\mathfrak q}^*\) (where \({\mathfrak q}^*\) is the dual of the Lie algebra \(\mathfrak q\) of \(Q\)) gives a separation of variables for the Hamiltonian system associated to \(H\). After reduction to \(K \times {\mathfrak k}^*\), the Hamiltonian equation in the cotangential variables becomes a so-called Lie-Poisson system. The author discusses some consequences of the invariance and symmetry assumptions and gives a condition for the momentum to be the constant of motion corresponding to the \(K\) symmetry of the Hamiltonian \(H\).
scientific article; zbMATH DE number 704245
scientific article