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

• Three symplectic realizations of the Lie–Poisson structure associated with Lie algebra sl(2) are presented.
• The notion can be used to study the relations among the restricted systems related to the same soliton hierarchy.
• The action-angle coordinates and Jacobi inversion problem for the Lie–Poisson Hamiltonian systems are also studied.
• The restricted Dirac system are used as an illustrative example.

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.