Quadratic Hamilton–Poisson systems on $\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}$: The homogeneous case

Abstract

In this paper we consider quadratic Hamilton–Poisson systems on the semi-Euclidean Lie–Poisson space [Formula: see text]. The homogeneous positive semidefinite systems are classified; there are exactly six equivalence classes. In each case, the stability nature of the equilibrium states is determined. Explicit expressions for the integral curves are found. A characterization of the equivalence classes, in terms of the equilibria, is identified. Finally, the relation of this work to optimal control is briefly discussed.