The Application Of The Lie-Poisson Structure Associated With The Symplectic Algebra Sp(2, R) In The Finite-dimensional Integrable Systems

In this paper, the Lie-Poisson structure associated with the 4 × 4 symplectic algebra is given, and the coupling KdV hierarchy is taken as an e?ective example to illustrate its application in the finite-dimensional integrable systems. Firstly, based on the Lie-Poisson structure associated with the 4 × 4 symplectic algebra, the coupling KdV nonlinearized eigenvalue problems are produced. Secondly, by making full use of the Lax representations in this structure and generating function approach, the integrability of the Hamilton system in the sense of Liouville is emphatically proved. Therefore, the application of the Lie-Poisson structure associated with the symplectic algebra in the finite-dimensional integrable systems is illustrated.