(Quasi) Hamiltonian Systems and Non-Decomposable Poisson Geometry
C. Sardón, X. Zhao
Abstract
In this work, we conduct a systematic study of Hamiltonian and quasi-Hamiltonian systems within the framework of nondecomposable generalized Poisson geometry. Our focus lies on the interplay between the algebraic structure of nondecomposable generalized Poisson brackets and the dynamical behavior of systems exhibiting specific symmetry properties. In particular, we demonstrate that if a dynamical system admits suitable invariance conditions -- such as those arising from Lie symmetries or conserved quantities -- it can be formulated as a quasi-Hamiltonian system, or even as a genuinely Hamiltonian system, with respect to a suitably constructed nondecomposable generalized Poisson structure. This result offers a unified geometric framework for analyzing such systems and underscores the capacity of nondecomposable generalized Poisson structures in contexts involving multi-Hamiltonian or higher-order dynamics.