Symplectic realizations and action-angle coordinates for the Lie-Poisson system of Dirac hierarchy

In this paper, the relations among four finite-dimensional Hamiltonian systems associated with the Dirac hierarchy are studied via three Poisson maps from the standard symplectic structures to the Lie-Poisson structure of Lie algebra sl(2). It has shown that the canonical Hamiltonian systems in the standard symplectic structures are the different symplectic realizations of the Lie-Poisson Hamiltonian system. The action-angle coordinates and the Jacobi inversion problem for the Lie-Poisson Hamiltonian systems are also investigated in detail. (C) 2014 Elsevier Inc. All rights reserved.