The Bargmann System And The Neumann System For The 4～(th) Order Eigenvalue Problems And The Representation Of The Solutions For The Evolution Equations

In this paper, By means of the Euler systems on the symplectic manifold, the Bargmann system and the Neumann system for the 4f/lorder eigenvalue problems:are gained. Then the Lax pairs for them are nonlinearized respectively under the Bargmann constraint and the Neumann constraint. By means of this and based on the Euler-Lagrange function and Legendre transformations, the reasonable Jacobi-Ostrogradsky coordinate systems are found, which can alsobe realized. Then the original systems can be transformed into Hamiltoniancanonical systems in the symplectic space Moreover, thesolutions for the equations are gained.